GRE Math : Right Triangles

Study concepts, example questions & explanations for GRE Math

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Example Questions

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Example Question #1 : How To Find An Angle In A Right Triangle

A triangle has three internal angles of 75, 60, and x. What is x? 

Possible Answers:

90

45

60

75

110

Correct answer:

45

Explanation:

The internal angles of a triangle must add up to 180. 180 - 75 -60= 45. 

Example Question #1 : How To Find The Area Of A Right Triangle

Quantitative Comparison

 Gre_quant_171_01

Column A

Area

 

Column B

Perimeter

 
 
Possible Answers:

Column B is greater

Column A and B are equal

Cannot be determined

Column A is greater

Correct answer:

Column A and B are equal

Explanation:

To find the perimeter, add up the sides, here 5 + 12 + 13 = 30. To find the area, multiply the two legs together and divide by 2, here (5 * 12)/2 = 30.

Example Question #1 : How To Find The Area Of A Right Triangle

Gre_quant_179_01

Given triangle ACE where B is the midpoint of AC, what is the area of triangle ABD?

Possible Answers:

48

24

96

72

Correct answer:

24

Explanation:

If B is a midpoint of AC, then we know AB is 12. Moreover, triangles ACE and ABD share angle DAB and have right angles which makes them similar triangles. Thus, their sides will all be proportional, and BD is 4. 1/2bh gives us 1/* 12 * 4, or 24.

Example Question #1291 : Gre Quantitative Reasoning

What is the area of a right triangle with hypotenuse of 13 and base of 12?

Possible Answers:

25

30

60

156

78

Correct answer:

30

Explanation:

Area = 1/2(base)(height). You could use Pythagorean theorem to find the height or, if you know the special right triangles, recognize the 5-12-13. The area = 1/2(12)(5) = 30. 

Example Question #101 : Geometry

Quantitative Comparison

Quantity A: the area of a right triangle with sides 10, 24, 26

Quantity B: twice the area of a right triangle with sides 5, 12, 13

Possible Answers:

The relationship cannot be determined from the information given.

Quantity A is greater.

The two quantities are equal.

Quantity B is greater.

Correct answer:

Quantity A is greater.

Explanation:

Quantity A: area = base * height / 2 = 10 * 24 / 2 = 120

Quantity B: 2 * area = 2 * base * height / 2 = base * height = 5 * 12 = 60

Therefore Quantity A is greater.

Example Question #1291 : Gre Quantitative Reasoning

Quantitative Comparison

Quantity A: The area of a triangle with a height of 6 and a base of 7

Quantity B: Half the area of a trapezoid with a height of 6, a base of 6, and another base of 10

Possible Answers:

The two quantities are equal.

The relationship cannot be determined from the information given.

Quantity A is greater.

Quantity B is greater.

Correct answer:

Quantity B is greater.

Explanation:

Quantity A: Area = 1/2 * b * h = 1/2 * 6 * 7 = 42/2 = 21

Quantity B: Area = 1/2 * (b1 + b2) * h = 1/2 * (6 + 10) * 6 = 48

                Half of the area = 48/2 = 24

Quantity B is greater.

Example Question #1 : Right Triangles

The radius of the circle is 2. What is the area of the shaded equilateral triangle?

Capture3

Possible Answers:

\dpi{100} \small 2\sqrt{2}

\dpi{100} \small \pi \sqrt{3}

\dpi{100} \small 3\pi

\dpi{100} \small 3\sqrt{3}

\dpi{100} \small \pi \sqrt{2}

Correct answer:

\dpi{100} \small 3\sqrt{3}

Explanation:

This is easier to see when the triangle is divided into six parts (blue). Each one contains an angle which is half of 120 degrees and contains a 90 degree angle. This means each triangle is a 30/60/90 triangle with its long side equal to the radius of the circle. Knowing that means that the height of each triangle is \dpi{100} \small \frac{r\sqrt{3}}{2} and the base is \dpi{100} \small \frac{r}{2}.

Applying \dpi{100} \small \frac{bh}{2} and multiplying by 6 gives \dpi{100} \small 3\sqrt{3}). 

Capture4

Example Question #1 : How To Find If Right Triangles Are Similar

Quantitative Comparison

Quantity A: The area of a triangle with a perimeter of 34

Quantity B: 30

Possible Answers:

The relationship cannot be determined from the information given.

Quantity A is greater.

The two quantities are equal.

Quantity B is greater.

Correct answer:

The relationship cannot be determined from the information given.

Explanation:

A triangle with a fixed perimeter does not have to have a fixed area. For example, a triangle with sides 3, 4, and 5 has a perimeter of 12 and an area of 6. A triangle with sides 4, 4, and 4 also has a perimeter of 12 but not an area of 6. Thus the answer cannot be determined.

Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

2 triangles are similar

Triangle 1 has sides  6, 8, 10

Triangle 2 has sides 5 , 3, x

find x

Possible Answers:

12

5

9

4

2

Correct answer:

4

Explanation:

Draw the triangles

Triangle 1 is a 6,8,10 right triangle with 10 as the hypotenuse

Triangle 2: 3 is half of 6, 5 is half of 10; x must be half of 8

Example Question #1 : How To Find The Length Of The Side Of A Right Triangle

Varsity_tutors_problem1

Given the diagram, indicate if Quantity A is larger, Quantity B is larger, if they are equal, or if there is not enough information given to determine the relationship. 

Quantity A: \dpi{100} \small x

Quantity B: 7.5

Possible Answers:

Quantity B is greater.

The two quantities are equal.

Quantity A is greater.

The relaionship cannot be determined from the information given.

Correct answer:

Quantity A is greater.

Explanation:

Since this is a 30-60-90 triangle, we know that the length of the side opposite the 60 degree angle is \dpi{100} \small \sqrt{3} times the side opposite the 30 degree angle. Thus, \dpi{100} \small 5\sqrt{3}, which is about 8.66. This is larger than 7.5. 

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