# Geometry

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GMAT Study Guide - a prep wikibook

## Lines

"Line" is a basic concept in geometry. It refers to a straight line extending in both directions.

### Intersecting lines, vertical angles

When two lines intersect, the angles created by these lines possess special qualities. The opposite angles are equal in measure. These opposite angles are also known as vertical angles. The sum of adjacent angles equals 180 degrees.

If the two lines intersect at right angle (90 degrees), these lines are perpendicular. All 4 angles, created by these lines are right angles.

### Parallel lines

The two lines are parallel if they don't intersect. Parallel lines never intersect, no matter how far they extend. When two parallel lines are intersected by the third line, two intersections with vertical angles are created.

## Polygons

Polygon is a plane figure consisting of 3 or more line segments connected to form a closed space. These line segments are named sides; points where the sides meet are named vertices.

Convex polygon is a polygon in which each of the interior agnles is less than $180^o$.

On GMAT, the most widely used kinds of polygons are triangles and quadrilaterals.

The sum of all interior angles of a polygon is found with this formula:

$\Large (n - 2) * 180^o$

Where $n$ is the number of vertices of a polygon. Thus, the sum of angles of a triangle equals $180^o$. Quadrilateral's sum of angles equals $360^o$.

The perimeter of a polygon is the sum of lengths of its sides.

The area of a polygon is the area enclosed within the sides of a polygon.

## Triangles

### Definition

Triangle is a polygon consisting of three vertices and three line segments connecting these vertices.

### Types of triangles

Equilateral triangle is a triangle with all three sides equal in length. All three angles of an equilateral triangle are equal to $60^o$.

Isosceles triangle is a triangle with two sides equal in length. The two angles opposite to the equal sides are equal as well.

Right triangle has one angle equal to $90^o$. Hypotenuse is the side of a right triangle opposite to the right angle. Other two sides are called catheti (singular - cathetus).

Obtuse triangle has one of the angles equal to more than $90^o$ (obtuse angle).

Acute triangle has one of the angles equal to less than $90^o$ (acute angle).

### Properties

• Sum of three angles of any triangle equals $180^o$.
• Sum of lengths of two sides always exceeds the length of the third side.

Two triangles are similar if angles of one triangle are equal to the corresponding angles of the other. Lengths of the corresponding sides of similar triangles are proportional.

### Area

Area of a triangle is generally computed as follows:

$S=\frac{1}{2}ah_a$, where $a$ is a side and $h_a$ is the height or altitude of a triangle dropped from the vertex opposite to side $a$.

$S=\sqrt{p(p-a)(p-b)(p-c)}$ - for all triangles ($p$ is a half-perimeter, $a$, $b$ and $c$ are lengths of the sides)

$S=\frac{a^2\sqrt{3}}{4}$ - for equilateral triangle with side $a$

$S=\frac{1}{2}ab$ - for right triangle with catheti $a$ and $b$

### Pythagorean theorem

A right triangle

Pythagorean theorem states that for any right triangle the square of length of the hypotenuse equals the sum of the squared lengths of the catheti. If $c$ is the hepotenuse and $a$ and $b$ are the catheti of the right triangle then:

$\Large a^2 + b^2 = c^2$

The backward statement is also true: if the sides of a triangle satisfy the given equation then the triangle is a right triangle.

There are some special right triangles which are worth remembering (it might save you some time on the test day):

• a 3-4-5 triangle is a right triangle with hypotenuse equal to 5 and catheti of 4 and 3 units respectively $(4^2 + 3^2 = 5^2)$. Note that you may encounter right triangles with sides which are multiples of 3, 4 and 5, respectively. The most common of them is a 6-8-10 triangle
• a 5-12-13 triangle has a hypotenuse equal to 13 and catheti of 5 and 12 respectively $(5^2 + 12^2 = 13^2)$.

In a right triangle with one of the angles equal to $30^o$ the shorter cathetus equals half the length of the hypotenuse.

Also note that in any right triangle the length of the median dropped to the hypotenuse is half the length of the hypotenuse.

### Lines of triangles

BD is a median of $\tiny\triangle$ABC. CD=AD. $\small S\tiny\triangle \small ABD=S \tiny\triangle \small CBD$

Median is a line segment that connects a vertex and the midpoint of the opposite side. Median divides a triangle into two smaller triangles of equal area.

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Quadrilateral is a four-sided polygon. It has four vertices and four sides. We'll consider the most popular quadrilaterals here. These are squares, rectangles, parallelograms, trapezoids, rhombuses.

Square ABCD.
AO=BO=CO=DO.

### Squares

Square is a quadrilateral with all sides and all angles equal. Each angle of a square measures $90^o$.

Square is also called regular quadrilateral.

#### Properties

• Opposite sides of a square are parallel
• All sides of a square are equal
• Diagonals of a square are equal
• Diagonals form a right angle and bisect each other
• Angle formed by a diagonal and an adjacent side equals $45^o$
• All squares can be inscribed into a circle (squares are cyclic quadrilaterals)

Area = $a^2$, where $a$ is the length of the side.

Perimeter = 4*a.

Length of a diagonal equals $\sqrt2*a$.

Parallelogram ABCD.
AB=CD.
AE=EC, BE=ED.

### Parallelograms

Parallelogram is a quadrilateral with opposite sides parallel and equal in length.

#### Properties

• Opposite sides of a parallelogram are equal and parallel
• Diagonals of a parallelogram are bisecting each other
• Opposite angles of a parallelogram are equal
• Two angles adjacent to the same side sum up to $180^o$

Area = $a*h_a$, where $a$ is the side and $h_a$ is the height drawn down to the side $a$.

Perimeter = $2a + 2b$, where $a$ and $b$ are different sides of parallelogram.

Rectangle ABCD.
AB=CD.
AC=BD.

### Rectangles

Rectangle is a parallelogram with right angles.

#### Properties

• Opposite sides of a rectangle are equal and parallel
• Diagonals of a rectangle are equal and bisecting each other
• All angles of a rectangle are right

Area = $a*b$, where $a$ and $b$ are different sides of a rectangle.

Perimeter = $2a + 2b$.

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Rhombus ABCD.
AK=KC, BK=KD.

### Rhombuses

Rhombus is a parallelogram with all sides equal.

#### Properties

• All sides are equal
• Diagonals of a rhombus intersect under right angle

Area = $a*h$, where $a$ is the side and $h$ is the height drawn down to the side $a$.

Area = $\frac{1}{2}*c*d$, where $c$ and $d$ are diagonals of a rhombus.

Perimeter = $4*a$.

Trapezoid ABCD.

### Trapezoids

Trapezoid is a quadrilateral with two opposite sides parallel.

#### Properties

• Two of the four sides are parallel

Area = $\frac{b+d}{2}*h$, where $b$ and $d$ are the parallel sides and $h$ is the height of the trapezoid.

Perimeter = $a + b + c + d$.

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## Circles

Circle with center O. Radius = OB = OA = OC. DC is the chord. AB is the diameter. Central angle COB is subtended by the arc CEB. Line AF is tangent to the circle. $\angle$OAF is right.

### Definitions

A circle is a set of points on a plane equidistant from a certain point (called the center of the circle).

A radius is a segment connecting the center of the circle and the point on the circle.

A chord is a segment connecting two points on the circle.

A diameter is a chord passing through the center of the circle.

An arc is any part of a circle.

A central angle of a circle is the angle whose vertex is the center of the circle and the sides pass through the two points on the circle.

A line is tangent to the circle if it has only one point common with the circle.

### Area and circumference

The circumference is a perimeter of a circle. It is equal to $2\pi r$, where $r$ is the radius of a circle. $\pi$ is approximately 3.14.

The area of a circle is equal to $\pi r^2$.

The length of an arc is equal to $\frac{x}{360}$ of the circumference of the circle, where $x$ is the central angle subtended by the endpoints of the arc. If $\angle$COB from the picture above equals 45 degrees, then the length of the arc CEB equals $\frac{45}{360}=\frac{1}{8}$ of the circumference of the circle.

$\angle$COB is the central angle.
$\angle$CDB = $\angle$CAB = $\frac{1}{2}\angle$COB.
$\angle$CEB + $\angle$CDB = $\small 180^\circ$.
$\angle$CAB = $\angle$CEB = $\angle$CDB = $\small 90^\circ$

### Angles

Here are some important properties of the inscribed angles:

• A central angle equals twice an inscribed angle if they are subtended by the same chord (if both angles are on the same side of the chord). The image on the left illustrates it, $\angle$COB = 2$\angle$CDB = 2$\angle$CAB.
• Two inscribed angles subtended by the same chord and on the same side of the chord are equal (on the left: $\angle$CAB = $\angle$CDB).
• Two inscribed angles subtended by the same chord and on the opposite sides of the chord are supplemental (the sum of the two angles equals $\normal 180^\circ$). See the image on the left: $\angle$CAB + $\angle$CEB = $\normal 180^\circ$.
• All inscribed angles subtended by a diameter equal $\normal 90^\circ$ (see the image on the right).

## Coordinate geometry

### Equation

Equation of a line is $y=mx+b$, where m=slope and b=y intercept.

Equation of a circle is $(x - a)^2 + (y-b)^2 = r^2$, where (a,b) is the center and $r$ is the radius.

Equation of a circle is $x^2+y^2=r^2$ if (0,0) is the center.

• Points that solve the equation of a line are in the same line
• Given a point and slope, equation of the line can be found
• Given the equation, x and y intercepts can be found

Intercept

• Y intercept is the value of y when x is 0
• X intercept is the value of x when y is 0

### Distance

Distance between two points = $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$.

### Midpoint

$Xm=\frac{x_1+x_2}{2}$

$Ym=\frac{y_1+y_2}{2}$

### Slope

Slope $m =\frac{(y_1-y_2)}{(x_1-x_2)}$.

• A straight line with a -ve slope passes through II and IV quadrants
• A straight line with a +ve slope passes through I and III quadrants
• If the slope is 1 the angle formed by the line is 45 degrees.
• If the slope of a line is n, the slope of a line perpendicular to it is its -ve reciprocal, -1/n.
• If a line is horizontal, slope=0, equation is y=b.
• If a line is vertical, slope is not defined, equation is x=a, where a is x-intercept.
• Parallel lines have same slope.