## CHAPTER SUMMARY

### I. Basics

Basics

Geometry questions on the GMAT are a logical puzzle. You will be given some information about a figure, and must deduce some piece of missing information. These questions test a set of general rules and formulae that must be memorized.

#### Lines

Through any two points, there is exactly one line, extending infinitely in both directions. A section of a line is called a line segment or a ray.

#### Angles

When two rays originate from the same point, they form an angle, which is represented by the symbol ∠. The point of intersection is the vertex and the two rays form the sides of the angle. Angles that have the same measure are congruent.

#### Types of Angles

Angles can be categorized by their angle measure: acute, right, obtuse, straight.

Two angles are complementary angles or complements if the sum of their measures is 90°. Two angles are supplementary angles or supplements if the sum of their measures is 180°.

#### Midpoint and Bisector

The midpoint is the center point of any line segment. Bisect means “to cut in half.” A bisector is a line that divides a line segment or an angle into two equal pieces.

### II. Intersecting Lines

Intersecting Lines

#### Vertical Angles

When two lines intersect, they create four angles. The angles opposite one another are called vertical angles.  Vertical angles are equal to each other. Also, note that any pair of adjacent angles will always equal 180°

#### Parallel Lines

Two lines that never get closer to or farther away from each other and therefore never intersect are called parallel lines.  The symbol for parallel lines is ||.  When two parallel lines are cut by a third line, called a transversal, the three lines form a system of congruent angles.

#### Perpendicular Lines

When two lines intersect each other at a right angle (90°), the lines are perpendicular.  The symbol for perpendicular is ⊥, and m ⊥ n means that line m and line n are perpendicular lines.

Be careful not to assume that lines are parallel or perpendicular if it’s not stated explicitly. Don’t be fooled by appearance; you cannot assume that figures are drawn to scale.

### III. Triangles

Triangles

There are some rules that are true for all the different kinds of triangles. Be sure to familiarize yourself with these rules.

#### Perimeter

The perimeter of any figure is the distance around the outside of the figure, or the sum of the sides of the figure. Many GMAT questions about triangle perimeters involve the rule that the length of the longest side of a triangle must be less than the sum of the lengths of the other two sides of the triangle, but greater than the difference of the lengths.

#### Area

The area of any figure is the amount of surface that is covered by the figure.

The formula for the area of a triangle is A = \dfrac{1}{2} (base × height) = \dfrac{1}{2} bh

#### Types of Triangles

There are three important types of triangle:

• An isosceles triangle has two equal sides and two equal angles.
• An equilateral triangle has three equal sides and three equal angles.
• right triangle is a triangle with a 90° angle.

#### Right Triangles

For right triangles, the relationship between the legs and the hypotenuse is defined by the Pythagorean Theorem. The Pythagorean Theorem states that the square of the hypotenuse will equal the sum of the squares of the legs.

#### Special Right Triangles Based on Side Lengths

There are several special right triangles where the lengths of their sides form ratios of whole numbers. These triangles appear commonly on the GMAT, so being able to quickly recognize them will help you solve problems more quickly and accurately.

#### Special Right Triangles Based on Angles

There are two special “angle-based” right triangles on the GMAT:

• The 45° : 45° : 90° triangle
• The 30° : 60° : 90° triangle

#### Similar Triangles

Triangles that have the same shape are called similar triangles.

There are two ways to know that two triangles are similar:

• If the corresponding angles are equal, the triangles are similar.
• If the corresponding sides have the same ratio, the triangles are similar.

The ratio of the lengths of the corresponding sides is called the scale factor.  The scale factor for congruent triangles is 1.

### IV. Polygons

Polygons

polygon is a figure made from 3 or more line segments. Triangles and quadrilaterals are the most common polygons on the GMAT. Other polygons on the test usually contain triangles and quadrilaterals.

#### Perimeter and Area

• The perimeter of a polygon is the distance around the outside of the polygon, or the sum of the lengths of the sides of the polygon.
• The area of a polygon is the amount of surface that is covered by the polygon.

Familiarize yourself with the formulas for finding area and perimeter of the different types of polygons.

#### Interior and Exterior Angles

The sum of the interior angles of a polygon is a function of the number of sides.  The formula for the sum of the interior angles of a polygon with n sides is (n – 2)180°.

The sum of the exterior angles of all polygons is 360°. You can think of this as going all the way around the polygon, making a complete circle.

#### Symmetry

A figure has line symmetry if the figure can evenly “fold” onto itself.  A figure can have more than one line of symmetry.

A figure has rotational symmetry if the figure can be rotated less than 180° and lines up on the original figure. The point used for the rotation is the center of symmetry.

#### Similar Polygons

Polygons that have the same shape but a different size are called similar polygons. Polygons are similar if the corresponding angles are equal, and the corresponding sides have the same ratio.

The ratio of the lengths of the corresponding sides is called the scale factor.  The scale factor for congruent figures is 1.

#### Diagonals

diagonal of a polygon is a segment that connects two non-adjacent vertices of the polygon.

Diagonals of a polygon form triangles inside the polygon. The GMAT often uses the triangles formed inside quadrilaterals by the diagonals.

#### Maximum and Minimum

The GMAT often asks for the maximum or minimum perimeter or area. Maximum area is when a triangle or quadrilateral has perpendicular sides and right angles. Squares and 45° : 45° : 90° triangles have the maximum area for the smallest perimeter.

### V. Circles

Circles

circle is the set of all points in a plane that are the same distance from the center. A circle has 360°.

#### Basics

A segment from the center of the circle to the circle is called the radius r.

A segment that goes across a circle through the center is called the diameter d. A diameter of a circle is twice the radius.

The circumference of a circle is the distance around the circle. The formula for the circumference of a circle is C = 2πr = πd.

The formula for the area of a circle is A = πr2.

#### Central Angles

central angle has its vertex at the center of a circle, so it is formed by two radii.

An arc is a piece of a circle. The measure of a central angle equals the measure of the intercepted arc.

sector is a “piece of a pie” created by a central angle and its arc.  A sector has perimeter and area.

#### Inscribed Angles

An inscribed angle has its vertex on the circle and is formed by two chords. The measure of an inscribed angle equals half the measure of the intercepted arc.

#### Inscribed Polygons

An inscribed polygon is formed by chords so it has its vertices on the circle.
The GMAT often uses inscribed polygons.  You need to be able to combine the information about circles, angles and polygons.

#### Tangents to Circles

tangent to a circle is a line that touches the circle in just one point. A tangent is perpendicular to a diameter.

Angles formed by a chord and a tangent are half the measure of the arc.

Tangents from a common exterior point are congruent. The measure of the angle formed by the tangents is half the difference between the measures of the two arcs.

### VI. Solids

Solids

#### Rectangular Prisms

rectangular prism, or rectangular solid, is basically a box. It has 6 sides. All the sides, or faces, are rectangles.  Opposite sides are congruent. A special rectangular prism is a cube. All the faces are squares.

The surface area, SA, is the sum of the areas of each pair of congruent sides.

To find the volume of a rectangular prism, multiply the three lengths.
V = length × width × height = lwh

#### Cylinders

cylinder is like a can.  It has circles as the top and base, and straight sides.

• SA = 2 circles + rectangle = 2(πr2) + 2πrh
• V = πr2h

#### Spheres

sphere is like a ball.  The formulas for circumference, surface and volume all use the radius.

• C = 2πr =  πd
• SA = 4πr2
• V = (\dfrac{4}{3}) πr3

### VII. Coordinate Geometry

Coordinate Geometry

coordinate plane is formed by the intersection of a horizontal number line, called the x­-axis, and a vertical number line, called the y-axis.  The number lines intersect at zero at the point called the origin.

#### Slope of a Line

The slope of a linem, is its slant.  The slope can be used to give the rate of change.

The formula to find the slope of the line between two points (x1y1) and (x2y2) is

slope = = \dfrac{\,rise\,}{run} = \dfrac{change\,\, in\,\, \textit{y}}{change\,\, in\,\, \textit{x}}

= \dfrac{{\textit{y}}_{\displaystyle{2}} \,-\, \textit{y}_{\displaystyle{1}}}{\textit{x}_{\displaystyle{2}} \,-\, \textit{x}_{\displaystyle{1}}}

#### Graphing a Line

A graph of the equation Ax + By = C is a line.

There are two ways to graph a line:

• Slope-intercept form, y = mx + b  where m is the slope and the point (0, b) is the -intercept where the line crosses the y-axis.
• Graph using the intercepts.

#### Writing the Equation of a Line

An equation of a line can be found from two points, or from one point and the slope.

For the point (x1y1) and the slope m, the point-slope formula is
y – y1 = m(x – x1).

To write an equation from two points (x1y1) and (x2y2), first find the slope of the line.

slope = = \dfrac{{\textit{y}}_{\displaystyle{2}} \,-\, \textit{y}_{\displaystyle{1}}}{\textit{x}_{\displaystyle{2}} \,-\, \textit{x}_{\displaystyle{1}}}

Then substitute the slope m and one of the points into the point-slope formula.

#### The Distance Between Two Points

The distance between two points can be found using the Pythagorean theorem.

(horizontal distance )2 + (vertical distance )2 = (distance)2

#### Transformations

There are 3 common transformations that move or change a figure on the coordinate plane:

• translation moves a point or figure some distance up or down, left or right.
• reflection acts like a mirror over a line of reflection.
• dilation makes a similar figure that is bigger or smaller.

## Sample Questions

#### Review

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6 questions with video explanations

100 seconds per question 