Quadrilaterals: Overview

Q42...What is the height of P from AB in the figure above? 20 25 30 35 50

E Correct. Remember that height of one point from another is the perpendicular distance from the first to the second point. The height of point P from AB is the perpendicular distance of P from AB. Based on the given figure the height of P = 20 + 30 = 50. Hence the height of P is 50 cm.

Q37...Four congruent equilateral triangles are combined to form a larger equilateral triangle, as shown below. If the height of the larger triangle is 1, then what is the height of a smaller triangle? 2/3 1/2 1/4 1/6 1/9

B Correct. Remember that congruent triangles have same angles and same sides. Remember that the height of a triangle is the shortest distance from one vertex of the triangle to the opposing side. Given that all four triangles inside the large triangle are congruent, they all have the same height. Since the height of the larger triangle is composed of two smaller triangles and measures 1 in length, each smaller triangles is half of that. Hence, 1/2.

Q41...What is CD in the figure above? √2 2 √6 2√2 4

B Correct. Use the Pythagorean theorem of find DB, and then the recycled 45:45:90 ratio to find hypotenuse DC. Using the Pythagorean theorem: 12 + DB2 = (√3)2 = 3 --> DB2 = 3 - 1 = 2 Take the square root from both sides of the equation to get: --> DB = √2. Since DB=BC (according to the figure), DBC is a right isosceles triangle, which means that its sides satisfy the 1:1:√2 recycled ratio. Therefore, hypotenuse CD = √2×DB = √2×√2 = (√2)2 = 2.

Q40...In the figure above AB and CD are parallel. If the height of A from CD is 8, what is the height of C from AB? 4 4√2 8 8√2 16

C Correct. Remember that the height of a triangle is the perpendicular distance from the base of the triangle to the tip. Given that the height of A from CD is 8, the height of C from AB is also 8 since the two heights are equal distance between parallel line segments AB and CD.

Quadrilateral means four-sided. Hence, this is the common name for any four-sided figure. The sum of the angles in any quadrilateral is 360°.

Some quadrilaterals have special qualities and special names namely, parallelogram, rectangle, square etc. GMAT quadrilateral problems may require you to calculate the area or perimeter of a quadrilateral, as well as verify your understanding of various concepts regarding those special quads. We'll talk about the different quadrilaterals' properties in the near future.

By now, we've talked about all the different quadrilaterals you're likely to encounter on the GMAT, so you should be able to answer this simple question: ABCD is a parallelogram. What does ABCD look like? Note that the reverse is NOT true - For example, as the only regular quadrilateral, a square must have four equal sides and four right angles, and so cannot be drawn as "rectangle" or a "parallelogram". To sum up: A quadrilateral may also be a parallelogram, or a rectangle, or a square. A parallelogram may also be a rectangle, or a square. A rectangle may also be a square. A square is a regular quadrilateral.

A rectangle and a square are merely special instances of a parallelogram: A rectangle is a parallelogram with four right angles. A square is a parallelogram with four right angles and all sides equal. In the same manner, a rectangle doesn't have to look like a "traditional" rectangle, but can also look like this: Square We call this important concept the evolution of quadrilaterals, illustrated in the following diagram. Unless specifically noted by the question, any given quadrilateral can be drawn in any of the forms "included" within it on the diagram. Remember this important concept when dealing with GMAT problems which do not supply any figures. Data sufficiency problems may also try to trick you into thinking that you know what quadrilateral ABCD looks like, when in fact there is more than one way to draw it.

Q35...In the figure below, what is the value of x? 55° 60° 65° 70° 75°

B Correct. Remember that the sum of the angles inside any triangle is 180°. Also, angles on a straight line sum up to 180°.Sum of all angles in a triangle is 180°. Also, sum of all angles on a straight line is 180°. Angles CED and AED are supplementary angles so angle AED = 180° - 120° = 60°. Based on this angle DAE = 180° - 90° - 60° = 30°. Hence, angle BCE = 180° - 90° - 30° = 60°. (Sum of all angles in a triangle is 180°).

Q39...In quadrilateral ABCD triangles ABC and ADC are not congruent. Quadrilateral ABCD may be which of the following? rectangle parallelogram Isosceles trapezoid rhombus (equilateral parallelogram) square

C Correct. An isosceles trapezoid has 2 sets of equal angles, as shown in the figure. When diagonal AC cuts the figure into two triangles, ∠ABC and ∠ADC are not equal. Therefore the triangles ABC and ADC cannot be congruent.

Q37...Rectangle ABCD is divided into five identical rectangles, as shown above. If the perimeter of ABCD is 132, what is the length of side BC? 24 27.5 30 33 36

C Correct. In order to find BC, it is necessary to find out how the perimeter of 132 is divided between the different sides. Since the five small rectangles are equal, the figure is basically composed of two types of lines: lengths (call those L), and widths (W). Thus, the required BC is composed of L+W. The issue is therefore finding the length of L and W. To do that, focus on equal sides AB and CD: AB is composed of 2L, and CD is composed of 3W. Thus, 2L = 3W. It is also possible to express the given perimeter in terms of L and W. Perimeter = AB + BC + CD + DA = 132 -> Perimeter = (2L) + (L+W) + (3W) + (L+W) = 4L+5W = 132. Isolate L from the AB = CD equation. L = 3W/2 Plug this into the new equation: 4L+5W= 4(3W/2) + 5W = 132 --> 12W/2 + 5W = 132 --> 6W + 5W = 132 --> 11W = 132 --> W = 12 Thus, L = 3×12/2 = 18. Finally, BC = 18+12 = 30.

Q38...An equilateral triangle and three squares are combined as shown above, forming a shape of area 48+4√3. What is the perimeter of the shape formed by the triangle and squares? 18 27 36 48 64

C Correct. Start by breaking down the area of the shape into the different components. The 4√3 doesn't look like the area of a square - test the assumption that this is the area of the equilateral triangle: 4×√3=a^2√3/4 Solve for a2: multiply both sides by 4 to reduce the denominator /:√3 (to isolate a2) Therefore, a=4. Since a=4 is also the side of the squares, the area of one square is a2=16, and all three squares together add up to 48. Total area of shape: 48+4√3, as given in the question. So our original assumption for 4√3 was correct - yippee!. Now answer the question: the perimeter of the shape is the sum of its outer sides - 9 sides, or 9a = 36.

Q36...In rectangle ABDE above, points F and C are on sides AE and BD respectively, and line segment FC is parallel to side AB. If DC=3, CG=8, and GB=10, what is the area of ABDE? 72 96 108 120 144

C Correct. The area of a rectangle is given in the formula Area = Length × Width. Triangles BCG and BDE form a "triangle within a triangle" figure which points towards triangle similarity. Use the two sides BG and CG of right triangle to find the third side BC, and from there the width BD. Then figure out the ratio between the similiar triangles, and use it to find the missing length DE. Copy the figure to your noteboard and mark the given lengths. Triangle BCG is a recycled right triangle of 3:4:5 multiplied times 2 or a 6:8:10 ratio. If BG=10 and CG is 8, then BC must equal 6. Therefore, width BD = BC+CD = 6+3 = 9. Now, focus on the similar triangles BCG and BDE: The ratio between BC and BD is 6:9, or 2:3. Since respective sides in similar triangles maintain the same fixed ratio, the ratio between CG and ED must also be 2:3. Since CG = 8, this means that ED must equal 12. Thus, the area of rectangle ABDE = length × width = 12 × 9 = 108.

Is quadrilateral ABCD a square? (1) AB=BC (2) ABCD is a rectangle.

C Correct. This is a DS Yes\No question. The issue is finding out whether ABCD is a square, in other words, whether all its angles are right and all its sides are equal. Stat.(1) only tells you that two sides of ABCD are equal. The others sides could be equal, shorter, or longer, and even if all sides are equal, you still know nothing about the angles. ABCD could be a square (giving an answer of 'yes'), or any other kind of quadrilateral (giving an answer of 'no'). Stat.(1)->Maybe->IS->BCE. Not all rectangles are squares, so Stat.(2) isn't much help. Although rectangles have four right angles, their width may be different from their height. Stat.(2)->Maybe->IS->CE. Together, Stat.(1+2) give you the answer, by telling you ABCD is a rectangle in which the height and width are equal (AB=BC). Since opposite sides are equal in a rectangle, this means all sides are equal and all angles are right angles. ABCD must be a square. Stat.(1+2)->Yes->S->C.

Diagonal BD divides quadrilateral ABCD into two triangles. Are these triangles congruent? (1) AD||BC (2) AB=CD

C Incorrect. Remember that one counter-example is enough to prove insufficiency of a statement or a set of statements. E Correct! The issue here is whether quadrilateral ABCD MUST be divided into two congruent triangles by diagonal BD. By finding even one quadrilateral which cannot be divided into two congruent triangles, you have proved that both statements are insufficient. If both statements are true, the quadrilateral is either an isosceles trapezoid (a trapezoid whose legs are of equal length) or a parallelogram. By definition, every parallelogram has congruent sides and therefore can be divided into 2 congruent triangles. However, an isosceles trapezoid meets conditions 1 and 2 but when divided by line AC, the resulting triangles are not congruent.

Mr. Green divided $360 evenly among his children. Jerry, the youngest, decided to give each of his brothers and sisters $4 of his share. Eventually, Jerry ended up with $8. How many children does Mr. Green have? 4 5 6 8 9

E Correct. Numbers in the answer choices and a specific question ("How many children...?") call for Plugging In The Answers. You may feel like writing down one equation or more. This is just your algebraic urge, which is another stop sign for Reverse PI problems. Assume the amount in each answer choice is the number of children among which $360 are divided and then follow the story in the problem. If everything fits - stop. Pick it. Otherwise - POE and move on, until you find an answer that works. Start with answer choice C. Assume that each child got $360/6=$60. Jerry then gives each of his 5 siblings $4, and eventually he has $60-(5x$4)=$40. This does not fit the information in the problem. Therefore POE C. You need an answer that would leave Jerry with $8, therefore, you need more siblings. Hence, POE A and B. Now, plug in D or E, to check which is correct. The correct answer is E: Assume that each child got $360/9=$40. Jerry then gives each of his 8 siblings $4, and eventually he has $40-(8x$4)=$8.

A parallelogram is a quadrilateral with opposite sides that are equal and parallel. The inside angles behave as would the angles between parallel lines; Opposite angles are equal and adjacent angles sum up to 180°.

The perimeter of a parallelogram is the sum of its sides. The area of a parallelogram is given in the formula Area=Base×Height, where the Height is the distance between any two parallel sides, and is perpendicular to the Base. Note that one definition of parallel lines is equidistant. The diagonals of a parallelogram bisect each other, but are NOT necessarily equal. (Note that squares and rectangles are special types of parallelograms that do have equal diagonals.)

A rectangle is a parallelogram, and therefore shares all the same properties of a parallelogram with four right angles. The dimensions of a rectangle are length and width, or l and w, respectively.

The perimeter of a rectangle is the sum of its sides, given by the formula Perimeter=2w+2l. The area of a rectangle is given in the formula Area=Length×Width. Since the sides are perpendicular to each other, they're analogous to Base and Height. The diagonals of a rectangle are equal and bisect each other.

A square is a rectangle, and therefore shares all the same properties of a rectangle with all sides equal.

The perimeter of a square is the sum of its sides, given by the formula Perimeter=4s, where s is the side. The area of a square is given in the formula Area=Side2. The diagonals of a square are equal and perpendicular, bisect each other, and bisect their respective base angles.

A Rhombus is a special instance of a Parallelogram with all sides equal. It is otherwise identical in all its properties to a Parallelogram, with the following additional property: The diagonals of a Rhombus are perpendicular to each other, as well as bisect each other and their respective base angles.

What is the perimeter of an equilateral triangle with height 2√3? 3 6√3 3+√3 6+2√3 12 E The perimeter of any triangle is the sum of its sides An equilateral triangle can be divided into two 30, 60, 90 triangles, when cut in half. Correct. Draw a median to split the equilateral triangle into two equal 30, 60, 90 triangles. Use the recycled 30:60:90 triangle to find the sides of the triangle. The resulting right triangles have sides 2, 2√3 and 4. Hence, the perimeter is 4×3=12.