Coordinate Systems
Q29... In the regular octagon above, what is x? 157.5° 150° 145° 135° 120°
D Good! Draw the octagon inside a circle. The octagon divides the circle into eight arcs, each of which measures 360°/8 or 45°. Each inscribed angle equals half the arc segment defined by its rays. Therefore, ∠BCA is half of arc segment BA, or 22.5°. ∠CBD equals half of arc segment DC, or 22.5°. Therefore ∠CMB equals 180-22.5-22.5=135°. Because x and ∠AMB are opposite angles, x=135°.
Q5...If a circle of diameter 18 is circumscribed around triangle ABC, as shown above, what is the length of arc ADC? 3 6 3π 18 6π
E Correct. The sum of angles in a triangle is 180, thus angle ABC = 180- (60+60) = 60. The three angles of the triangle are identical inscribed angles, lying together on the entire circumference of the circle. Thus, Each of the three arcs in the figure is equal to 120°, or 1/3 of the circumference (i.e. 1/3 of 360, the sum of angles in a circle). Now that you know that the arc is 1/3 of the circumference, find the circumference. If the diameter is 18, the circumference = πd = 18π. Thus, arc ABC = (1/3)18π = 6π.
The y-intercept is where a line intersects the y-axis. The coordinates of the y-intercept are always (0,y), thus for y-intercept x=0. In the example above, point (0,-2) is the y-intercept.
The x-intercept is where a line intersects the x-axis. The coordinates of the x-intercept are always (x,0), thus for x-intercept y=0. In the example above, point (4,0) is the x-intercept.
Parallel lines or equidistant lines are straight lines that never meet. Their distance from one another is always the same.
In the coordinate system parallel lines share the same slope. It doesn't mean that they're the same line.
Consider the following example, What is the slope of the line that passes through the points (4,1) and (2,5)? -2 -1/2 0 1/2 2
A Correct. Plug the numbers into the formula to get . 5-1 / 2-4 = -2 It doesn't matter which coordinate you label y1 or y2, as long as you use the corresponding x1 and x2.
Q24...In the regular hexagon above, what is x? 100° 108° 120° 135° 144°
C Good! Draw the hexagon inside a circle. The hexagon divides the circle into six arcs, each of which measures 360°/6 or 60°. Each inscribed angle equals half the arc segment defined by its rays. Therefore, ∠FEA is half of arc segment FA, or 30°. ∠EFD equals half of arc segment ED, or 30°. Therefore ∠EMF equals 120°. Because x and ∠EMF are opposite angles, x=120°.
Q31...What is the length of line segment AB in the figure above? 5 6 5√2 6√2 10
C Correct. Draw a line parallel to the y-axis from B and a line parallel to the x-axis from A till the two lines intersect to form a triangle. Apply the 45-45-90 triangle ratio to find the length of AB. The base and height of the resulting triangle are (yB-yA)=3- (-2) and (xB-xA)=2- (-3), or 5 and 5. Hence, the hypotenuse of the triangle, AB, is 5√2.
The slope of line m is −1. If line m passes through point (9,4), what are the coordinates of the x-intercept of line m? (0,13) (0,−5) (13,0) (5,0) (−5,0)
A Incorrect. The slope is given in the formula: The x-intercept is where a line intersects the x-axis at the point (x,0), thus for x-intercept y=0. This answer choice can be eliminated on sight, because its y coordinate is not zero. Don't make careless mistakes - use POE. D Incorrect. C Correct. Use the slope formula using the two points (9,4) and (x,0), where x is the x coordinate of the x-intercept. (4-0) / (9-x) = -1. Multiply by 9-x to get 4=-(9-x)=x-9 Therefore, x=4+9=13.
Q36...If O is the center of the circle shown above, what is the value of x? 100º 110º 120° 130º 140º
C Correct. This question is involves arcs and angles. Use the method you have learned: 1. Copy the figure to your noteboard. 2. Indicate the measure of EVERY arc on the circle. 3. Then solve for x. This is what your noteboard should look like: The central 120° angle defines an arc of 120°, marked in Red. The measure of the remaining part of the circumference, i.e., the Blue arc, is 360°−120°=240°. Now, solve for x: The 240° arc is twice the inscribed angle that defines it, therefore x is half of 240: 240°/2=120°.
Q30...If E is a point on the circumference of the circle circumscribing square ABCD, as shown above, what is the value of x? 45 90 120 135 270
D Correct. COPY the figure, INDICATE the measure of every arc next to it: The square's vertices divide the circle's circumference (360°) into 4 equal arcs of 90° each. The angle x is an inscribed angle that defines an arc that is made of three 90° arcs. Hence, x defines a 3×90° arc. Recall that the arc is twice the inscribed angle that defines it, therefore the angle x is half the 3×90° arc: --> x = 3×45° = 135°
Q25...Points A and B and the origin are the three vertices of an isosceles triangle, as shown above. If the coordinates of A are (0,10), what are the coordinates of B? (0,10) (10,10) (−10,0) (0,−10) (10,0)
E This question can be solved very easily through POE. Point B lies on the x axis, so its y coordinate=0. POE answer choices A, B and D. Since point B lies on to the right of the x axis, its x coordinate must be positive. POE Answer choice C, and you're left with E. Always try to POE - you never know how much time and energy you might save.
The coordinate plane (also known as the rectangular coordinate system) is a two-dimensional plane with two perpendicular axes: x and y. The x-axis indicates the horizontal direction while the y-axis indicates the vertical direction of the plane.
The intersection of the axes is the origin. Each point on the coordinate plane can be described using its position along the x and y axes, relative to the origin. Points are generally notated with a capital letter.
Slope is used to describe the steepness of a straight line. A greater slope value indicates a steeper incline.
The slope is defined as the "rise" divided by the "run" between any two points on a line, in other words, the ratio of the vertical change to the horizontal change between any two points on the line. The slope is given in the formula: Slope = Vertical change / Horizontal Change = y2-y1/x2-x1 Remember: You need two different points on the line to get its slope.
Q23...In the polygon inscribed in circle O above, ∠A=90°. If AB=AE and BC=CD=DE, as shown above, what is the value of x? 100 105 115 120 150
B Correct. COPY the figure, INDICATE the measure of every arc next to it: Right Angle EAB defines a semicircle of 180°. Arc EDCB is divided into three equal arcs, each 60°. Arc EAB (which is also 180°) is divided into two equal arcs, each 90°. Now, x defines an arc of (60+60+90)°=210°. Recall that the arc is twice the inscribed angle that defines it, therefore the angle x is half of 210° i.e., x=105°
To summarize: Line Intersection
When lines in a coordinate system intersect, they have the same x and y at the point of intersection. To find the intersection point of two lines, equate their equations in the form y = mx+b.
Q7...If OB=AB=5 and OA=6, what is the slope of the line passing through A and B? 4/3 5/6 4/5 2/3 −5/3
A Correct. A 5:5:6 isosceles triangle can be split into two 3:4:5 right triangles. Remember to use recycled right triangle ratios to solve triangle questions. The slope is given in the formula Also, remember that the slope of an upward sloping line is always positive. Look at the triangle closely to identify that it can be split into two 3:4:5 recycled right triangles. Draw a straight line joining point B to the midpoint of OA. This line splits triangle OAB into two right triangles with a base of 3 and a hypotenuse of 5. Using the recycled ratio 3:4:5, the length of the line you have drawn is 4 i.e. the coordinates of point B are (3, -4). Use the coordinates of B and A to calculate the slope of line AB i.e. (0 - (-4)) / (6 - 3) = 4/3. Hence, the slope of line AB is 4/3. Alternative explanation: Ballpark the slope - Notice that if a line parallel to the x axis is drawn such that point B lies on that line, the angle between the new line and AB will be more than 45°. Whenever an upward sloping line makes an angle greater than 45°, the slope of the upward sloping line is greater than 1. You can eliminate all the other answer options using the above short cut.
Q11...If in the figure above, square S2 has side of length 2 and square S1 has side of length 1, what is the equation of line m? y=(1/3)·x+1 y=(1/2)·x+1 y=(1/3)·x y=(1/2)·x y=x−1
A Correct. Notice that the side of square S1 is 1. Hence, the y-intercept of line m is 1. Eliminate all answer options without a y-intercept of 1. Find the slope of m to choose between the remaining answer options. The slope (rise/run) of line m is 1/3 because difference between the heights of S1 and S2 is 1 (rise = 2 - 1 = 1) and the sum of their sides is 3 (run = 2 + 1 = 3). Hence, the slope is 1/3.
Q20...In the figure above, what is the distance from point P to point Q? √2 1.52−1 √(1.52−1) (√2)/2 0.5
A Incorrect. Drop a line parallel to the y-axis from Q and a line parallel to the x-axis from P till the two lines intersect to form a triangle. Apply the 45-45-90 triangle ratio to find the length PQ. C Incorrect. B Incorrect. D Since the base of the triangle is parallel to the x-axis and the height is parallel to the y-axis, the coordinates of the third vertex of the triangle are (1.5, 1). Hence, the base and height of the resulting triangle are 0.5 (x2 - x1 = 1.5 - 1 = 0.5) and 0.5 (y2 - y1 = 1.5 - 1 = 0.5). Apply the 45-45-90 (1, 1, √2) triangle ratio to find the length PQ. Hence, the hypotenuse of the triangle PQ is 0.5√2 = (1/2)√2 = √2/2.
Q13...In the regular pentagon above, what is x? 108° 112° 120° 126° 144°
A Nice work! Draw the pentagon inside a circle. The pentagon divides the circle into five arcs, each of which measures 360°/5 or 72°. Each inscribed angle equals half the arc segment defined by its rays. Therefore, ∠MAB is half of arc segment BC, or 36°. ∠MBA equals half of arc segment EA, or 36°. Therefore ∠AMB equals 180°-36°-36°=108°. Because x and ∠AMB are opposite angles, x=108°.
A line equation or linear equation is an algebraic expression of a line in the coordinate system. Most of the Coordinate Geometry problems in the GMAT concern line equations.
A line equation has the form: y=mx+b. x and y are variables standing for the coordinates of any point on the line. m is the slope of the line. b is the y coordinate in which the line intersects the y-axis (where x=o). This point is also termed y-intercept.
To sum up: Slope is used to describe the steepness of a straight line. A greater slope value indicates a steeper incline You need two different points on the line to get its slope. It doesn't matter which coordinate you label y1 or y2, as long as you use the corresponding x1 and x2. The slope is given in the formula: The slope can be positive, negative, or zero. Ballparking slopes:
A rising line with slope 1 creates a 45° angle with the x axis. Therefore, a rising line with a degree greater than 45° will have a slope greater than 1.e.g. slope=2 A rising line with a degree smaller than 45° will have a positive slope smaller than 1. e.g slope = 1/2 In the same manner, A declining line with slope -1 creates a 45° angle with the x axis. Therefore, a declining line with a downward degree greater than 45° will have a slope smaller than -1. e.g. slope = -2 A declining line with a downward degree smaller than 45° will have a negative slope greater than -1. e.g. slope = -1/2
Q6...If quadrilateral ABCD shown above is a square, what is the slope of the line that passes through B and D? −2 −1 −0.5 0.5 1
B Correct. The slope is given in the formula Remember all four sides of a square are equal. Also, the downward sloping diagonal of a square has slope (-1) and the upward sloping diagonal of a square has slope (1), if the sides of the square are parallel to the y-axis. Since side DC lies on the y axis, all the sides of ABCD are parallel to the axes X and Y. The vertical change and the horizontal change are the same, hence the slope is 1. In fact, since it's a downward slope, it's (-1). The slope of the downward sloping diagonal of a square (with sides parallel to the y-axis) is -1, since it forms a 45 degree angle with the axis. If you're not sure, plug in your own numbers for the sides of the square and use the slope formula. Plug in numbers for the side of the square and the vertices. For example, Let's assume that point D is of coordinates (0, -1), and that the side of the square is 2. Thus, point C will be of coordinates (0, -3) (two down from D), and point B will be of coordinates (2, -3) (move two to the right from point C. Use these plug ins to calculate the slope of the line between B and D: -3 - (-1) / 2-0 = -2 / 2 = -1.
Now, you give it a try: Which of the following coordinates describes the intersection point of the linear equations y = -x+4 and y = 2x+7? (-1,3) (-1,5) (1,3) (2,-1) (5,-1)
B Correct. First set the two equations equal to each other: -x+4 = 2x+7 --> -x+4 (-2x) = 2x+7 (-2x) --> -3x+4 (-4) = 7 (-4) --> -3x (/-3) = 3 (/-3) --> x = -1 Then substitute the obtained x value into one of the original equations to solve for y: y = -(-1) +4 --> y = 1+4 --> y = 5 Therefore, the intersection point is (-1,5).
Q15...If B is the center of the circle in the figure above and the area of the shaded region is 16π, what is the length of arc ADC? 4 4π 16 8π 16π
B Correct. In order to find the length of an arc in a circle, you need two pieces of data: the circle's radius and a central or inscribed angle intercepting the said arc. Here you have central angle ABC = 90º. 90/360 = 1/4, and thus the area of sector ABC is 1/4 of the area of the circle: Now plug in the area of the sector and the central angle to find the circle area: 16π/circle area = 90/360 = 1/4 --> circle area = 4·16π = 64π Now you can find the circle's radius: circle area = πr2 = 64π --> r2 = 64 --> r = 8 Now, find the circle's circumference: --> 2πr = 2·8·π = 16π Arc ADC is 1/4 of the circumference = 4π
Q32...If O is the center of the circle shown above and BC=CD, what is the value of x? 15° 20° 25° 30° 40°
B Correct. In ΔACD, ∠ACD is an inscribed angle intercepting the diameter of the circle and is therefore 90°. ∠CAD=180-(70+90)=20°. Hence, minor arc CD is 40°. Minor arcs BC and CD lie on identical chords and are therefore equal. Thus, minor arc BC is 40°, and x=20°.
Q2...In the rectangular coordinate system above, the line y=x is the perpendicular bisector of line segment PQ (not shown.) If the coordinates of point P are (4,1), what are the coordinates of point Q? (−4,−1) (1,4) (−4,1) (−1,4) (1,−2)
B Correct. Rather than performing multiple calculations, use visual ballparking: If the line y=x is the perpendicular bisector of line PQ, then point Q must lie on the other side of line y=x, and be at an equal distance from it as point P. Draw the figure in your notebook, and place point Q in the appropriate place. It is obvious that Q must be in the first quadrant. POE all answer choices that do not fit this description. This answer choice is the only one with positive x and y coordinates, which is necessary for a point Q in the first quadrant.
Which of the following equations describes a line that is perpendicular to y=x+6? y = x − 6 y = −1 − x y = 6x y = 6x+1 y = 1/x + 6
B Correct. The equation of a line is y=mx + b, where m is the slope of the line and b is the y-intercept. Also, the slope of a line perpendicular to a line with slope m is -1/m. A line perpendicular to the line y=x+6 must have a slope of -1 because the slope of y=x+6 is 1. The slope of y=-1-x i.e. y=-x-1 is -1. Hence, this is the correct answer.
Q12...In the figure above, line segment QP lies on a line whose equation is which of the following? y=7/6 x y=x+1 y=7/4x + 1 y=2/3x + 1 y=1/2x + 1
B Correct. Use the coordinates and lengths given in the question to find the coordinates of points P and Q. Then calculate the slope of the line PQ using these coordinates. Eliminate all answer choices with a slope other than the one calculated for line PQ. Alternative method: Draw a line parallel to the x-axis from point P. The resulting triangle is an isosceles right triangle., or 45:45:90 triangle. Therefore, angle P is a 45 degree angle, and the slope of a line with an angle of 45 to the x axis is 1. Eliminate all answer options without a slope of 1. Given that the length of the line joining P to the x-axis is 3, the coordinates of P are (2, 0+3) i.e. (2, 3). Likewise, given that the length of the line joining Q to the x-axis is 7, the coordinates of Q are (6, 0+7) i.e. (6, 7). Hence, the slope of line PQ, m = (7 - 3) / (6 - 2) m = 4 / 4 m = 1 Since no other answer choice has a slope of 1, this is the correct answer. Alternative method: Draw a line parallel to the x-axis from point P, as shown in the figure below. The resulting triangle is an isosceles right triangle. Notice that the acute angle between PQ and the line you drew is 45°. The upward slope of the hypotenuse of this triangle is 1. Hence, this is the correct answer.
Q34...In the regular pentagon above, what is x? 76° 72° 60° 48° 45°
B Great! Draw the pentagon inside a circle. The pentagon divides the circle into five arcs, each of which measures 360°/5 or 72°, so the measure of arc DCB (the arc defined by x) is 72⋅2=144. Now, none of the answer choices equals 144, but remember that x is an inscribed angle. Each inscribed inscribed angle equals half of the measure of the arc defined by its rays. Thus ∠BAD, or x, is half of the measure of arc segments BC+CD, or 144/2 = 72°.
Which of the following points does NOT lie on the line y=−1.5x? (3,−4.5) (0,0) (−2,−3) (−3,4.5) (10,−15)
B Incorrect. Notice that x and y coordinates cannot be both positive or negative for the same point. For the line y=-1.5x, y is negative when x is positive and vice versa. Use this to eliminate answer options. Plugging in (0, 0) satisfies the given equation of the line. 0 = -1.5(0). Hence, this is not the correct answer. C Correct. Plugging in (-2,-3) does not satisfy the given equation of the line. -3≠ -1.5(-2) = 3. Hence, this point cannot lie on the given line. Alternative method: C is the only answer choice for which both x and y are negative. However, based on the equation y=-1.5x, x and y cannot both be negative. Therefore, this is the right answer choice.
Q19...The equation of line m is y=x/2+1. What is the distance from the x-intercept of m to the intersection of x=4 and m? √(13) 5 6 3√5 7
B Incorrect. Remember, the y-coordinate of the x-intercept is always 0. Find the y-coordinate of the point where x=4 for this line by plugging in 4 in the equation y = (x/2) + 1. Drop a line from this point to the x-axis to make a triangle. Calculate the distance from the x-intercept of m to the intersection of x=4 and m using the Pythagorean theorem. C Incorrect. A Incorrect. D Correct. Plug in y=0 to find the x-intercept of y = (x/2) + 1. 0 = (x/2) + 1 --> -1 = x/2 --> x = -2 Find the y-coordinate of the point where x=4 for this line by plugging in 4 in the equation y = (x/2) + 1. y = (4/2) + 1 = 2 + 1 = 3. Drop a line from the point (4, 3) to the x-axis to make a triangle as shown in the figure. The base and height of the resulting triangle are 6 ({4 -(-2)} = 4 + 2 = 6) and 3 (3 - 0 = 3). Use the Pythagorean theorem to calculate the hypotenuse of the triangle. 3² + 6² = √hypotenuse, so hypotenuse = √(9 + 36) = √45 = √(9 x 5) = 3√5. Hence, this is the correct answer.
If ABC is an equilateral triangle, AC lies on the x-axis, and A is closer to the origin than C, what are the coordinates of point A? (1) The coordinates of point C are (6,0). (2) The coordinates of point B are (4,2√3).
B You slightly overestimated the time this question took you. You actually solved it in 1 minutes and 31 seconds. Good. This is a "what is the value of..." DS question. In this type of question, a statement will be sufficient only if it leads to a single value of the variable (or expression) you're asked about. Since A is on the x-axis, you already know its y coordinate -- 0. The issue is finding A's x coordinate. Statement (1) gives you C's x coordinate, but you have no means of knowing the length of AC, i.e., the distance between A and C. The side of the equilateral triangle can be of any length, placing A anywhere between 0 and C on the x-axis. Stat.(1)->IS->BCE Now turn to statement (2). Remember the Recycled Right Triangles? Draw the height AD to form a 30-60-90 triangle. Point D's coordinates are (4,0), and the length of BD is 2√3 (since BD is perpendicular to the x axis, its length is its y coordinate). In a recycled 30-60-90 triangle the length of one side is sufficient to find the other two, so you can find that AD=2 using the recycled ratio box. Using this fact and D's x coordinate, you can calculate A's x coordinate. Stat (2)->S->B.
Sometimes, the question asks about the length of a line that's not parallel to any of the axes. What is the distance between point A and point C? What should you do next? 1..Subtract the x coordinate. 2..Draw lines connecting the x and y coordinates of A and C all the way to the axes. 3..Subtract the y coordinate.
C Correct. Having done that, connect the points A and C. Line AC isn't parallel to any of the axes. Form a right triangle and inspect it. What should you do next? Use the pythagorean theorem. Use the coordinate plane distance formula. Use a recycled right triangle. C Great choice. Recycled right triangles are real time savers in the GMAT. In this case, the 6:8:10 recycled right triangle is the triangle of choice. It is actually the 3:4:5 triangle, multiplied by 2. Also, remember that in order to identify simple recycled right triangles, you need only two sides of the triangle. And so, if the legs of the right triangle are 6 and 8, the hypotenuse must be 10 (the 6:8:10 triangle). The length of AC is 10. No calculations needed.
Q10...In the rectangular coordinate system above, the line x=0 is the perpendicular bisector of line segment AB (not shown). If the coordinates of point A are (4,−1), what are the coordinates of point B? (4,1) (−4,1) (−4,−1) (−1,4) (0,−1)
C Correct. Remember, line x=0 is actually the y-axis. Rather than performing multiple calculations, use visual ballparking: If the line y axis is the perpendicular bisector of line AB, then point B must lie on the other side of y axis and be at an equal distance from it as point A. Draw the figure in your notebook, and place point B in the appropriate place. It is obvious that B must be in the third quadrant. POE all answer choices that do not fit this description. This is the only answer choice with both x and y coordinates negative, which fits the description of B in the third quadrant.
Q20... If in the figure above, the area of the triangle AOB is 30 and the y-intercept of line l is 5, what is the x-intercept of line l? 25 24 12 6 0
C Correct. The coordinates of the y-intercept and x-intercept of a line are always (0, y) and (x, 0) respectively, where y and x are the distances of the two points from the origin. Given that the y-intercept of the line is 5, the length of line AO is 5. Plug in this value into the area of a triangle formula i.e. 30 = (1/2) x OB x 5. 30 x 2 = 5 x OB 12 = OB. Hence, the x-intercept of line AB is 12.
Lines y=√3·x−2 and y=2√3·x−5 intersect at what height above the x axis? 0 1/(√3) 1 √3 5
C Correct. The height above the x-axis at which the two points intersect is the y-coordinate of the point of intersection. In order to find the point of intersection equate the two equations. Equate the two equations to find the point of intersection i.e.√3(x)-2 = 2√3(x)-5 √3(x)-2√3(x) = -5+2 √3(x-2x) = -3 √3(-x) = -3 -x√3 = -3; divide both sides by -√3 x = √3 Based on this, x = √3. Plug this value of x in either equation to find the height i.e. the y-coordinate of the point of intersection. y = √3(x)-2 = √3√3 - 2 = 3 - 2 = 1. Hence, this is the correct answer.
Q18...If the distance from P to O in the figure above is k, then what is the distance from O to point Q (not shown,) whose coordinates are (4x, 4y)? k/2 2k 4k 8k 16k
C Correct. Variables in the question and answer choices? Plug in good numbers and POE. Draw a line joining P to the x-axis such that the line makes a right angle. Plug in your own values for x and y. It will save you time to plug in x=4 and y=3 to get hypotenuse=5 using the 3:4:5 recycled ratio. Multiply the values of x and y by 4 and find the hypotenuse of the resulting triangle. Based on the figure above, length of line OP i.e. k is 5 because Since the 3:4:5 ratio extends to multiple of the same ratio, Multiply OP i.e. k by 4 to get the length of OQ i.e. OQ = 5 x 4 = 20. That's your Goal. Now plug k=5 into the answer choices and POE those which do not match your goal. Since no other answer choice matches your Goal of 20 for k=5, this is the right answer choice.
Q21...If the circle in the figure above is centered at the origin of the coordinate axes, which of the following coordinates represents a point that lies on the circle? (3,4) (5,5) (1,9) (8,6) (6,6)
C Incorrect. Eliminate answer choices with coordinates that surely lie inside the circle rather than on the circumference. Find the distance of the remaining answer choices from the center of the circle by drawing right triangles i.e. connecting the point given in each answer choice with the x and y axes using lines parallel to the two axes. The hypotenuse of the triangle is the distance of the point from origin, which should be equal to the radius. Use the Pythagoras theorem to find the hypotenuse and eliminate all choices with a hypotenuse other than 10 as the circle shown in the figure has radius 10. Based on the figure above, the height and the base of the right triangle are 9 and 1 respectively. Using the Pythagoras theorem hypotenuse2 = 92 + 12 hypotenuse2 = 81 + 1 hypotenuse2 = 82 hypotenuse = √82 Since √82 is not equal to 10, this is not the correct answer. D Correct. Based on the figure above, the height and the base of the right triangle are 6 and 8 respectively. Using the Pythagoras theorem hypotenuse2 = 82 + 62 hypotenuse2 = 64 + 36 hypotenuse2 = 100 hypotenuse = √100 hypotenuse = 10 Since in this case the hypotenuse is equal to the radius 10, this is the correct answer.
When lines in a coordinate system intersect, for a brief moment they share the same coordinates. In other words, they have the same x and y at the point of intersection.
Consider the following example: What is the intersection point of the linear equations y = -2x+10 and y = x-2? Since the lines share the same coordinates at the point of intersection, equate the linear equations like so: -2x+10 = x-2; Then solve: --> -2x+10 (-x) = x-2 (-x) --> -3x+10 (-10) = -2 (-10) --> -3x (/-3) = -12 (/-3) --> x = 4 Thus the x-coordinate is 4. To find the y-coordinate, plug in the x-value using one of the linear equations. In the example above, plug x = 4 into the linear equation y = x-2 to get: y = (4)-2 --> y = 2 The point of intersection of y = -2x+10 and y = x-2 is (4,2).
Q26...In the figure above, ABCO is a rectangle and the line that passes through points A and B is 3y−2x−15=0. What is the equation of the line that passes through points O and C? y=3/2x - 7.5 y=2x - 5 y=3/2x y=2/3x y=2/3x - 15
D Correct. The equation of a line is y = mx + c, where m is the slope of the line and c is the y intercept. Parallel lines have the same slope. Lines AB and OC have the same slope because they are parallel lines. Make y the subject of the equation given in the question to find the slope of line AB. 3y−2x−15 = 0 or 3y = 2x + 15 or y = (2x + 15)/3 or y = (2/3)x + 5. Based on this, the slope of line OC is 2/3 and the y-intercept of line OC is 0 (according to the figure). Hence, the equation of line OC is y = (2/3)x.
Q37...What is the length of line segment PQ in the figure above? √10 5 4√2 2√13 10
D Correct. Draw a line joining P to the x-axis so that the line makes a right angle. Use the Pythagorean theorem to calculate the length PQ from the triangle formed as a result of connecting P and the x-axis. Apply the Pythagorean theorem to the base and height to get the hypotenuse PQ. The length of the leg lying on the x axis is 4-(-2) = 6. The length of the leg parallel to the y axis is (0-(-4)) = 4. Therefore, (PQ)² = 6² + 4² = 36 + 16 = 52. Hence, PQ = √52 = √(13 x 4) = 2√13.
Q22...In the figure above OA is perpendicular to OB. The coordinates of point B may be which of the following? (−4,2) (4,−2) (2,4) (4,2) (0.5,0.5)
D Correct. The slope of a line is given by . Remember that the vertical change is the change of the height of the line when moving to the right. The slopes of perpendicular lines in a coordinate system are reverse and reciprocal. Find the slope of line OA. The vertical change is 4 and the horizontal change is −2, giving it a slope of −4/2. The slopes of perpendicular lines in a coordinate system are reverse and reciprocal, so your goal is a line OB whose slope is 2/4, or 1/2 reduced. For each answer choice, find the slope of OB, using the slope formula - see if it matches 1/2. Do some POE first - Answer choices A and B are negative, while the figure shows point B in the first quadrant. Answer choice E can also be POEd because if Bs coordinates were (0.5, 0.5), Bs vertical and horizontal change would be equal, and point B would then be at 45 degree to the origin. Answer choice D is correct because it the slope of OB is 2/4: 2-0 / 4-0 = 1/2
Q9...The line 3x+2y=4 passes through all of the quadrants in the coordinate plane except: I II III IV II and IV
D Incorrect. Plug in numbers into the line equation to find the coordinate of two points on the line. Then plot the line on the coordinate system and see which quadrant the line does not pass through. C Correct. Plug in numbers into the line equation to find the coordinate of two points on the line. Then plot the line on the coordinate system and see which quadrant the line does not pass through. If x=0, 2y=4, or y=2. Therefore, the point (0, 2) is on the line, as it fulfills the line equation. If y=0, 3x=4, or x= 4/3. Therefore, the point (4/3, 0) is on the line, as it fulfills the line equation. Now draw the coordinate system and plot the line between the two points: The line does not pass through quadrant III.
When Tom works alone he chops 2 lb. salad in 3 minutes, and when Tammy works alone she chops 3 lb. salad in 2 minutes. They start working together, and after some time finish chopping 65 lb. of salad. Of those 65 lb., the salad quantity chopped by Tammy is what percent greater than the quantity chopped by Tom? 44% 100% 125% 225% 400%
D Incorrect. This problem looks like a combined rate problem: find the individual rate of Tom and Tammy, add to get their combined rate, find the time it takes them to chop 65 lb. of salad, find the individual amounts chopped by each one, then figure out the difference in percents. However, while the method described above will work, it's the long way to go about solving this question. There really is no need to find the time and the individual amounts of each one, since the problem asks merely for the ratio between the amounts. Since Tom and Tammy work for the same time, these amounts are determined by their respective rates only; find the rate ratio, and you have the work ratio. To illustrate what we mean by this, figure out the quantity of salad chopped by both choppers over the same common time (for example, 6 minutes): Tom chops 2 lbs. in 3 minutes, so in 6 minutes he will chop 2×2=4 lbs. Tammy chops 3 lb. in 2 minutes, so in the same 6 minutes she will chop 3×3=9 lbs. The quantity ratio is therefore 4 to 9, and this ratio is maintained whether they work for 6 minutes, 10 minutes, or 60 minutes (try it!). This is why finding the time it takes them to cut 65 lb. of salad is irrelevant. All you have to do now is answer the question - 9 is what percent greater than 4? This is the right answer for the wrong question. If the question were phrased as follows: The quantity chopped by Tammy is what percent of the quantity chopped by Tom?", then 9 is indeed 225% of 4. However, the presence of the word "greater" means that the question asks about the difference between the two, as a percent of Tom's quantity. C Correct Use the percent change formula: . (9-4)/4 ·100 = 5/4 ·100 = 125%.
The coordinates of point P are (x,y). If the distance of P from the origin of the axes is 7, which of the following represents a point whose distance from the origin is not 7? (0,−7) (y,x) (−y,x) (2x,2y) (−x,−y)
D Variables in the answer choices? Make the algebra go away - plug in good numbers and eliminate. Correct. As the coordinates are doubled the distance changes. Plug in (0, 7) for (x, y). The distance between (0, 14) and (0, 0) i.e. origin is not 7. Hence, this is the correct answer.
Q4...Which of the following represents the coordinates of a point that is inside the rectangle above? (2,2) (-4,5) (−4,4) (−2,−5) (−4,−1)
E Correct. A point INSIDE the rectangle has: --> an x value larger than -5 and smaller than 2 --> a y value larger than -2 and smaller than 4 --> x=-4 is larger than -5 and smaller than 2 and y=-1 is larger than -2 and smaller than 4. Thus, the point is inside the rectangle.
Q21...Which of the following is NOT an equation of one of the boundary lines of the shaded square in the figure above? y=x+1 y=x−1 y=−x+1 y=−x+3 y=2x−1
E Correct. Notice that the lines joining points (0, 1) and (1, 2), and (1, 0) and (2, 1) have slope 1. Also, the lines joining points (0, 1) and (1, 0), and (1, 2) and (2, 1) have slope -1. POE all answer options with a slope of 1 or -1. None of the lines surrounding the shaded square have a slope of 2. Hence, this is the correct answer.
Q33...If the coordinates of points A and C are (−4,2) and (1,−1), what is the area of the rectangle ABCD shown above? 3 5 9 12 15
E Correct. Remember the area of a rectangle is Area=Length×Width. Use the coordinates of the two points given to calculate the width and length of the rectangle shown in the figure. Since the sides of ABCD are parallel to the axes, the y coordinate of B is the same as that of A and the x coordinate of B is the same as that of C i.e. the (x, y) coordinates of B are (1, 2). Likewise, the y coordinate of D is the same as that of C and the x coordinate of D is the same as that of A i.e. the (x, y) coordinates of D are (-4, -1). Based on this, the length of line AB is 1 - (-4) = 5 the length of line BC 2 - (-1) = 3. Hence, the area of rectangle ABCD = 3 x 5 = 15.
Q3...If the lengths of line segments OA and OB are 5 and 13 respectively, what are the coordinates of point B in the figure above? (13,5) (12,13) (12,−5) (5,12) (−12,−5)
E Correct. This question can be solved very easily by POE. According to the figure, point B lies in the third quadrant, with negative x and y coordinate. Only answer choice E fits, so there's no need to even resort to the 5:12:13 recycled ratio to find that the x coordinate of B is -12. Always try to POE - you never know how much time and energy it may save you.
Q8...Which of the following is NOT an equation of one of the boundary lines of the shaded region in the figure above? x=5 y=0 y=2−2x x=0 y=2−x
E Incorrect. This answer choice presents a downward sloping line. The only downward sloping boundary line is the one joining points (2, 0) and (0, 2).Check if these two points satisfy the line equation: Plug in x=2 into the equation to get y=2-2=0. The point (2,0) is a point on the line in this answer choice. Plug in x=0 into the equation to get y=2-0=2. The point (0,2) is a point on the line in this answer choice. since only one line can pass between two points, the line with equation y = 2 - x is one of the boundaries of the shaded region - Eliminate this answer choice. B Incorrect. Eliminate this answer option because the line y=0 is the x axis, which is definitely one of the boundary lines - it's the line joining points (5, 0) and (2, 0). D Incorrect. Eliminate this answer option because the line x=0 is the y axis, which is definitely one of the boundary lines - it's the line joining points (0, 5) and (0, 2). C Correct. This answer choice presents a downward sloping line. The only downward sloping boundary line is the one joining points (2, 0) and (0, 2). Plug in x=2 into the equation to get y=2-2·2=2-4=-2. Since y does not result in zero, the point (2, 0) is not on the line in this answer choice. Therefore, The line with equation y = 2 - 2x is not one of the boundaries of the shaded region, and this is the correct answer.
Q14...If P is the center of the circle shown above, and BAC=30º, and the area of triangle ABC is √3, what is the radius of the circle? 1/2 1/(√2) 1 √2 2
E Incorrect. You're asked to find the radius of the circle. ΔABC is a recycled 30:60:90 triangle - ABC is an inscribed angle lying on the diameter and therefore it is a right angle, and BAC = 30º. The area of ΔABC = ½×BASE×HEIGHT = ½×AB×BC. The side ratio in this triangle is x:x√3:2x. Note: If AC, the diameter, which is also the triangle's hypotenuse is 2r, then the little leg, BC, is r and AB is r√3. The triangle's area = ½×BASE×HEIGHT = ½×AB×BC = ½×r√3×r=√3. r²=2 -> r=√2 D Correct. Using the recycled ratio, if AC, the diameter, which is also the triangle's hypotenuse is 2r, then the little leg, BC, is r and AB is r√3. The triangle's area = ½×BASE×HEIGHT = ½×AB×BC = ½×r√3×r=½×√3×r2 = √3. Isolate r2: multiply by 2 and divide √3: --> r2=2 --> r=√2
A negative slope means that the line declines from left to right: A positive slope means that the line rises from left to right: A horizontal line has a slope of zero, whereas a vertical line has no slope at all (undefined):
One final note: The slope of a line can be ballparked visually. A rising line with slope 1 creates a 45° angle with the x axis. Therefore, a rising line with a degree greater than 45° will have a slope greater than 1.e.g. slope=2 A rising line with a degree smaller than 45° will have a positive slope smaller than 1. e.g slope = 1/2 In the same manner, A declining line with slope -1 creates a 45° angle with the x axis. Therefore, a declining line with a downward degree greater than 45° will have a slope smaller than -1. e.g. slope = -2 A declining line with a downward degree smaller than 45° will have a negative slope greater than -1. e.g. slope = -1/2 Remember this concept - some coordinate system problems will allow a visual POE using ballparking.
Q16...Some coordinate geometry problems require you to find distances between two points on the coordinate plane. What is the distance between point A and point B? What is the distance between point B and point C?
Start by marking down the x and y coordinates all the way to the axes. Connect the points to see that AB parallels the x-axis (A and B have the same y coordinate), and BC parallels the y-axis (B and C have the same x coordinate). ?Ask a tutor Line segment AB extends horizontally from 1 to 9: subtract its x coordinates to find the length 9-1=8. Line segment BC extends vertically from 2 to 8: subtract its y coordinates to find the length 8-2=6.
An ordered pair refers to two objects, one of which is known as the first coordinate and the other as the second coordinate. The notation for an ordered pair is (a, b), with first coordinate a and second coordinate b. An ordered pair refers to two objects, notated as (x, y), with first coordinate x and second coordinate y. An ordered pair (x,y) may refer to a point in a coordinate system. An ordered pair (x,y) may also describe the solution set for simultaneous equations.
The pair is "ordered" in the sense that (a, b) is different from (b, a), unless a and b are the same. In the GMAT, an ordered pair (x,y) may refer to a point on a coordinate system, such that the first object, x refers to the x-coordinate, and the second object, y refers to the y-coordinate. An ordered pair may also describe the solution set for simultaneous equations. The solution set for 3x-4y=17 and 6x-4y=26 is x=3 and y=-2. The solution set may be written as the ordered pair (3,-2). Thus, the first object refers to the x value and the second object refers to the y value.
To sum up:Determining Lengths in The Coordinate Plane
To find the length of a line in the coordinate system: If the line is parallel to the axes, subtract the coordinate values of its endpoints. e.g. a line drawn between A(1,8) and B (9,8) is simply of length 9-1 = 8. If the line isn't parallel to the axes, draw a right triangle, then look for a right recycled triangle. If you can't find any, use the Pythagorean theorem to find the required sides. e.g. the line AC in the figure below is of length 10, calculated by drawing a recycled right triangle 6:8:10.
The slopes of two lines that are perpendicular to each other are reverse reciprocal:
if the slope of one line is m, then the slope of the other is -1/m. The slopes of perpendicular lines are reverse and reciprocal. In the above example the slope of one line is 2/5, so the slope of the perpendicular line is the reverse reciprocal = -1 / 2/5 = -5/2