# Triangle Geometry Review, Practice Problems with Solutions for GMAT

Updated on October 15, 2012 Our topic in this hub is to review triangle geometry concepts, rules and formulas and then cover 6 practice questions to excel on the subject to get ready any question comes in this area in GMAT exam.

NOTE: Due to limitation to display line segment symbol over line in html, line segment will be only named as their start and end point letter.

## TRIANGLE REVIEW

A triangle is a polygon with 3 vertices and 3 sides. Triangle on the right is denoted as ΔABC. A, B and C are vertices. AB, AC and BC are sides of the triangle.

Area (1/2) * BC * h

Perimeter AB + BC + AC

## Rule 1

The sum of interior angle of a triangle is 180ο

x + y + z = 180ο

## Rule 2

Any side of the triangle length is always less than sum of the length of other sides.

AC < AB + BC

AB < AC+ BC

BC < AB + AC

## Rule 3

All sides and angles of an Equilateral Triangle are equal. All angles are 60ο

<A = <B = <C = 60ο

AB = BC = AC

## Rule 4

An isosceles triangle has 2 interior angles equal measure and opposite side of those angles are same length.

<y = <z

AB = AC

## Rule 5

If a triangle has a right angle, it is called right triangle.

y = 90ο

Hypotenuse Rule AB2 + BC2 = AC2

Known whole number length for right triangle are
32 + 42 = 52 (3:4:5 ratio)

52+ 122 = 132 (5:12:13 ratio)

Area ABC = (1/2) * AB * BC

When angles of a right triangles are 45ο : 45ο : 90ο sides ratio will be 1:1:√2

When angles of a right triangles are 30ο : 60ο : 90ο sides ratio will be 1:√3:2

## PROBLEM SOLVING PRACTICE QUESTIONS

1) ΔABC is an isosceles triangle AB=AC. What is x terms of y ?

a) 180 - y

b) 90 + y

c) 90 - y

d) 180 - 2y

e) 60 + y

Solution 1: The sum of inner angles of an triangle is 180ο.

x+y+z = 180

since y=z in isosceles triangle we can replace z with y from above equation

x + y + y = 180

x + 2y = 180

x = 180 -2y

## Figure for Practice Question 2

2) In ΔABC triangle on the right, if x : y ratio is 1 : 4 then x = ?

a) 15

b) 20

c) 22.5

d) 32.5

e) 35

Solution 2: From the ratio 4x = y.

The sum of inner angles of an triangle is 180ο.

x + 3x + 4x = 180

8x = 180

x = 180/8

x = 22.5

## Figure for Practice Solution 3

3) ΔABC is an equilateral triangle and the length of the side is 4. What is the area of the triangle?

a) 8

b) 12

c) 6√3

d) 4√3

e) 8√2

Solution 3: If we draw the height line, we end up special 2 right triangle. Each right triangle has special 1:√3:2 ratio.

See the figure right side.

Area = (1/2) * BC * h = (1/2) * 4 * 2√3 =4√3

## Data Sufficiency Practice Questions

1) What is the area of a triangle?

(1) The sides are 3, 4 and 5

(2) One of the vertices is 90ο

Solution 1:

Statement (1) requires such triangle to be an right triangle. 3 and 4 must be the length and 5 is the hypotenuse. Multiplication of legs and divide by 2 will give the area of the triangle.

Statement (2) without knowing Statement (1) is not sufficient to know the area of the triangle.

## Figure for Data Sufficiency Question 2

2) BCD is straight line. What is X=?

(1) a = 50ο

(2) AB = AC

Solution 2: Statement (1) alone is not sufficient to find out X.

Statement (2) alone is not sufficient to find out X.

Let us try to use both information together. If we know a=50ο and triangle is isosceles then Angle B and C is equal and (180 - 50)/2=65 since 65+x = 180. We can find X as 115ο

## Figure for Data Sufficiency Question 3

3) ΔABC is a triangle and the length of side BC is whole number.

What is the length of BC ?

(1) Angle B= 90ο

(2) Angle A is smaller than angle B and bigger than angle C

Solution 3:

Statement 1 requires the triangle to be an right triangle. by using formula we can find BC= 4

Based on Statement (2) 3 < BC < 5

BC is equal to 4.

Both Statements are alone is sufficient to solve the problem.

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