Ch. 17 Geometry
17.4.20 Similar triangles
If one triangle is simply an enlargement of another triangle, the triangles are similar. There are three general ways in which two triangles can be similar. Triangles are similar if: (1) Three angles of one triangle are the same measure as three angles of another triangle (2) The three pairs of corresponding sides have lengths in the same ratio (3) An angle of one triangle is the same measure as an angle of another triangle and the sides surrounding these angles are in the same ratio. Remember, If two triangles are similar, then their sides measurements should be in the same ratio as each other.
17.9 The maximum area of a rectangle
The maximum area of a rectangle Given a rectangle with a fixed perimeter, the rectangle with the maximum area is a square. A square will always have an area greater than that of a rectangle with an equal perimeter. Additionally, a circle provides the largest area given an equal perimeter. The minimum perimeter of a rectangle Given a rectangle with a fixed area, the rectangle with the minimum perimeter is a square.
17.4.1 Interior angles of a triangle
The measures of the three angles (or interior angles) of any triangle add up to 180°. In an equilateral triangle, each angle measures 60 degrees. The largest angle of a triangle is always opposite the longest side of the triangle; the smallest angle is opposite the shortest side. Equal sides will always be opposite equal angles.
17.12 The interior angles of a polygon
The sum of the interior angles of any polygon can be calculated by means of a simple formula. Important! Sum of the interior angles of a polygon = (n - 2) × 180, where n = the number of sides in the polygon. One interior angle in a regular polygon = (180(n−2))/n, where n is the number of sides of the polygon.
17.21.3 Surface area of a rectangular solid or cube
The surface area of a cube = 6s^2, where s is the length of one side (or edge) of the cube. The surface area of a rectangular solid = 2(LW) + 2(LH) + 2(HW), where L is the length, W is the width, and H is the height.
17.4.5 The area of a triangle
To calculate the area of a triangle, we need two pieces of information: 1) the base of the triangle and 2) the height of the triangle. The area of any triangle equals one-half of the product of the base of the triangle multiplied by the height of the triangle: Area = (Base x Height) / 2 Remember, the base of the triangle is always perpendicular to the height. If a question does not explicitly demonstrate this, then we do not yet know the height!
17.17.1 Triangles inscribed in a circle
When a triangle is inscribed in a circle, if one side of the triangle is also the diameter of the circle, then the triangle is a right triangle with the 90° angle opposite the diameter. Conversely, if a right triangle is inscribed in a circle, then the hypotenuse of the right triangle is a diameter of the circle.
17.21 The cube and the rectangular solid
A cube is a three-dimensional shape with six square sides, referred to as faces. Because the six faces that bound the cube are each squares, the length of any edge of a cube equals the length of any other edge. In addition, the area of any of the cube's faces equals that of any other face. The volume of a cube = (length edge)^3. The volume of a rectangular solid = length × width × height.
17.22 The right circular cylinder
A right circular cylinder is a cylinder whose bases are circles that are perpendicular to the height of the cylinder. The right circular cylinder is just a special case of cylinders in general, many of which are neither "right" nor "circular." The volume of a right circular cylinder = πr^2h. Surface area of a right circular cylinder = 2(πr^2)+2(πrh).
17.8 Squares
A square has four equal sides and four equal angles. Each angle measures 90 degrees. The two diagonals of a square are perpendicular to each other and bisect each other. Each diagonal divides the square into two 45-45-90 right triangles. We can easily calculate the length of a square's diagonal, d, using the formula: d=s√2 Area of a Square = Side^2 Perimeter of a Square = 4 × Side
17.4.17 The 30-60-90 right triangle
A 30-60-90 right triangle has a 30 degree angle, a 60 degree angle, and a 90 degree angle. The sides of a 30-60-90 triangle are in a ratio of x : x√3 : 2x, where: X represents the length of the side opposite the 30 degree angle x3√ represents the length of the side opposite the 60 degree angle 2x represents the length of the side opposite the 90 degree angle. Essentially, since the sides of a 30-60-90 right triangle always exhibit this same ratio of x: x3√: 2x, if we know the length of any one side of the triangle, we can determine the lengths of the other sides of the triangle. Review example 32 on 17.4.17 for a problem that tests this concept.
17.4.11 Pythagorean triples
A Pythagorean triple refers to the lengths of the three sides of a right triangle when each side is of integer length. The two triples that show up the most frequently on the GMAT are the 3-4-5 and 5-12-13 triples The 3-4-5 right triangle If the lengths of the two shorter sides of a right triangle are 3 and 4, then the length of the hypotenuse must be 5. As long as the ratio of the sides is 3:4:5, with 5 being the hypotenuse, the triangle is a 3-4-5 right triangle. Therefore, triangles with sides such as 6-8-10, 12-16-20, and 18-24-30 are also considered 3-4-5 right triangles since they all reduce to the same ratio. Remember, these only work if the hypotenuse (longest side opposite of the 90 degree angle) is the longest variable. If a right triangle has a hypotenuse of length 4 and a shorter side of length 3, this does not mean that the other shorter side has a length of 5. The 5-12-13 right triangle If the lengths of the two shorter sides of a right triangle are 5 and 12, then the length of the hypotenuse must be 13. Additionally, triangles with sides such as 10-24-26 and 15-36-39 are also considered 3-4-5 right triangles since they all reduce to the same ratio. Memorize these Pythagorean triples!
17.1 Lines, rays, and line segments
A line can be viewed as a geometric figure connecting two points and extending indefinitely from each point while keeping the line's straightness. Thus, all lines are straight; they have no curvature. If a line extends from only one of its points, then it is called a ray or half-line. If a line ends at both points, that is, it doesn't extend in either direction, it is called a line segment or simply a segment. These points would then be called endpoints. See the attachment for an example of this. While lines and line segments can be characterized by either direction AB or BA, rays can only begin with the end point and extend outward. Essentially, they can only exist as AB, not BA.
17.7 Rectangle
A rectangle is any quadrilateral with four right angles. The longer side of the rectangle is usually called the length; the shorter side is usually called the width. Area = Length x Width Perimeter = (2)Length + (2)Width The longest line segment that can be drawn within a rectangle is a diagonal. The diagonal of any rectangle is equal to √(Length^2 + Width^2)
17.11 Trapezoids
A trapezoid is a quadrilateral in which one pair of opposite sides are parallel but the other pair of opposite sides are not parallel. The parallel sides, which are never of equal length, are called the bases of the trapezoid. If the two nonparallel sides are equal in length, the trapezoid is referred to as an isosceles trapezoid. Area: The area of a trapezoid is calculated by adding together the lengths of the two parallel sides (known as the bases), multiplying their sum by the height, and dividing that product by two. Area of Trapezoid = ((Base1+Base2)*H)/2
17.2 Angles
An angle is a union of two rays sharing a common endpoint. This common endpoint is called the vertex of the angle. The two rays are called the sides of the angle. An acute angle is an angle whose measure is more than 0° but less than 90°. A right angle is an angle whose measure is exactly 90° An obtuse angle is an angle whose measure is more than 90° but less than 180°. A straight angle is an angle whose measure is exactly 180°. Angles are supplementary if their measures add up to 180°. Another way of saying this is that supplemental angles form a straight angle.
17.4.3 The exterior angles of a triangle
An exterior angle is the angle created by one side of a triangle and the extension of an adjacent side. In the diagram attached, angles x, y, and z are exterior angles. Given any polygon, when taking one exterior angle per vertex, the sum of the measures of the exterior angles will always equal 360°. Some GMAT problems will provide MORE than one exterior angle per vertex (corner). Remember that given only 3 vertices, sum of the angles must be 180 degrees. Be on the lookout for angles outside of a triangle which are not true exterior angles.
17.4.4 An exterior angle of a triangle is equal to the sum of the two remote interior angles
An exterior angle of a triangle is equal to the sum of its two remote interior angles. See the attachment for an illustration of this. Angle a is an exterior angle of the triangle. Angles x, y, and z are interior angles. In addition, angles x and y are remote, or on the opposite side of the triangle, from angle a. Thus, angles x and y are termed remote interior angles relative to angle a.
17.4.14 The isosceles right triangle (45-45-90 right triangle)
An isosceles triangle has two equal sides and two equal angles. In an isosceles right triangle, the two equal angles each measure 45 degrees; the third angle measures 90 degrees. This triangle is also referred to as a 45-45-90 right triangle. The area of a right triangle can be found using the formula: Area = (Length^2)/2, where l is the length of one of the equal sides. Remember, the sides of a 45-45-90 right triangle are in a set ratio of x: x: x√2, where x√2 represents the length of the hypotenuse and x represents the length of each of the shorter legs. If we know the length of any side of an isosceles right triangle, we can easily determine the lengths of all the other sides since the ratio between the sides doesn't vary. For example, if the length of one leg is 7 meters, then the length of the other leg must also be 7 meters, and the length of the hypotenuse must be 7√2 meters. Additionally, the area of a 45-45-90 right triangle is one-half of the area of a square. The important implication of this is that if we know the length of any side of a square, we can determine the length of the diagonal of the square (and vice versa). The diagonal of the square is the hypotenuse of both 45-45-90 triangles.
17.21.2 The longest line segment that can be drawn within a rectangular solid or cube
Given any rectangular solid or cube, the longest line segment that can be drawn within the solid will be a diagonal that goes from a corner of the box or cube, through the center of the box or cube, to the opposite corner. To find the longest line segment that can be drawn within a rectangular solid, use the extended Pythagorean theorem: d2=l2+w2+h2. When the solid is a cube, use d=s√3, where s is the length of one side (edge) of the cube.
17.4.9 The Pythagorean Theorem
If we have a right triangle, and we know the length of two of the sides of the triangle, we can use the Pythagorean theorem to calculate the length of the third side. It is important to remember that in a right triangle, the hypotenuse, or the side opposite the right angle, will always be the longest side of the triangle In any right triangle, A^2 +B^2 = C^2 Conversely, if given any triangle with sides A, B, and C, if C2 = A2 + B2, then the angle opposite side C must measure 90°, and thus the triangle must be a right triangle.
17.4.18 The equilateral triangle
In an equilateral triangle, all of the angles are the same measure (60 degrees), and all of the sides are the same length. Remember that within a given triangle, equal sides correspond to equal angles. The area of an equilateral triangle can be calculated by using the formula A = (s^2 * √3) / 4, where s is the length of one side of the triangle. Memorize this formula! Keep in mind that cutting an equilateral triangle in half forms two 30-60-90 triangles.
17.4.7 The triangle inequality theorem
In any triangle, the sum of the lengths of any two sides of the triangle is greater than the length of the third side, and the difference of lengths of any two sides of the triangle is less than the length of the third side. That is, if we have three sides, A, B, and C, it is true that A + B > C, that C + B > A, and that A + C > B. In addition, the difference of lengths of any two sides of the triangle is less than the length of the third side. That is, if we have three sides, A, B, and C, it is true that A - B < C, that C - B < A, and that A - C < B.
17.18 Shaded regions
In general, when a geometrical figure has both shaded and unshaded regions, the area of the entire figure - the area of the unshaded region = the area of the shaded region.
17.2.7 Parallel lines intersected by a transversal
In the diagram below, line one is parallel to line two, and both lines are cut by the transversal, t. A transversal is simply a line that passes through two or more lines at different points. See the attachment for reference. Vertical angles are angles that are diagonally oriented to each other. When lines are cut by a transversal, vertical angles are equal. A pair of corresponding angles are two non-adjacent angles that are on the same side of the transversal, but one is inside the parallel lines while the other is outside the parallel lines. When two parallel lines are cut by a transversal, corresponding angles are equal. A pair of supplementary angles can always be "pasted" together to form a straight line. When parallel lines are cut by a non-perpendicular transversal, both obtuse and acute angles are necessarily created. Two angles that are supplementary must add up to 180°. If the intersecting line is perpendicular to the parallel, then the resulting supplementary angles would both equal 90 degrees (half of a straight angle).
17.6 The parallelogram
Quadrilaterals, four-sided polygons, include squares, rectangles, parallelograms, rhombuses, and trapezoids. All squares are rectangles, and all rectangles are parallelograms. The parallelogram A parallelogram is a quadrilateral that has two pairs of parallel sides. Every parallelogram has the following features: Opposite sides are equal in length, opposite angles are equal in measure, the diagonals bisect each other, and each diagonal divides the parallelogram into two congruent triangles. The area of a parallelogram is found by multiplying its base by its height. The height is always perpendicular to the base. Area = Base x Height
17.13.1 The area of a regular hexagon
The area of a regular hexagon = (3√3)/2 *s^2, where s is the length of any of the hexagon's sides. A regular hexagon can be divided into six equilateral triangles
17.16 Circles
The circumference of a circle can be expressed as either C=2πr or C=πd. The area of a circle = πr2. A central angle is any angle at the center of the circle that is formed by two radii. An arc is a portion of the circumference of a circle. A sector of a circle is the region of a circle that is defined by two radii and their intercepted arc. Three-part circle ratio: = central angle / 360 = arc length / circumference = area of sector / area of circle The Minor Arc of a Circle: If points A and B are two points on a circle and arc AB is not a semicircle, arc AB refers to the shorter portion of the circumference between A and B. This shorter portion is also known as the minor arc. The longer portion of the circumference between A and B is known as the major arc but is usually referred as arc ACB, C representing some third point on this longer portion of the circumference. Inscribed Angles: The degree measure of an inscribed angle is equal to half of the degree measure of the arc that it intercepts. Essentially, Inscribed Angle degree = 1/2 arc angle
17.17.7 Squares inscribed in circles
When a square is inscribed in a circle, a diagonal of the square is also a diameter of the circle.
17.17.4 Equilateral triangles inscribed in circles
When an equilateral triangle is inscribed in a circle, the center of the triangle coincides with the center of the circle. If we were to draw a line segment from this center to a vertex of the triangle, not only would that line segment be a radius of the circle, but it would also bisect the 60 degree angle. When an equilateral triangle is inscribed in a circle, the triangle divides the circumference of the circle into three arcs of equal length.
17.2.4 Intersecting lines
When n lines intersect through a common point, the sum of all the angles created by those n lines at that point is 360 degrees. When intersecting lines are perpendicular, each angle formed between those lines is 90°. Another way of saying this is that two straight lines are said to be perpendicular when they intersect and form four right angles. Remember that a right angle measures 90°.
