GRE Math : Plane Geometry

Study concepts, example questions & explanations for GRE Math

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Example Questions

Example Question #1 : How To Find The Perimeter Of An Acute / Obtuse Isosceles Triangle

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An acute Isosceles triangle has two sides with length  and one side length . The length of side   inches. If the length of  , what is the perimeter of the triangle? 

Possible Answers:

 inches

 inches

 inches

 inches

Correct answer:

 inches

Explanation:

In order to solve this problem, first find the length of the missing sides. Then apply the formula: 

The missing side equals:



Then plug each side length into the perimeter formula:



 

Example Question #2 : How To Find The Perimeter Of An Acute / Obtuse Isosceles Triangle

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An acute Isosceles triangle has two sides with length  and one side length . The length of side   inches. If the length of , what is the perimeter of the triangle?

Possible Answers:

 feet

 feet

 inches

 inches

 inches

Correct answer:

 inches

Explanation:

In order to solve this problem, first find the length of the missing sides. Then apply the formula: 

The missing side equals:





Then, apply the perimeter formula by plugging in the side values: 

Example Question #1 : How To Find The Perimeter Of An Acute / Obtuse Isosceles Triangle

An acute Isosceles triangle has two sides with length  and one side length . The length of side   ft. If the length of   , what is the perimeter of the triangle? 

Possible Answers:

 foot

 foot

 foot

 inches 

 inches

Correct answer:

 inches 

Explanation:

To solve this problem apply the formula: .

However, first calculate the length of the missing side by:

 , Note that 




Since, it takes  inches to make one foot, the perimeter is equal to  inches. 

Example Question #4 : How To Find The Perimeter Of An Acute / Obtuse Isosceles Triangle

An acute Isosceles triangle has two sides with length  and one side length . The length of side   inches. The length of side  . Find the perimeter of the triangle.

Possible Answers:

 inches

 inches

 inches

 inches

 inches

Correct answer:

 inches

Explanation:

To find the perimeter of this triangle, apply the formula: 









Note: Since this is an acute Isosceles triangle, the length of the base must be smaller than the length of both of the equivalent sides. 

Example Question #11 : Triangles

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The obtuse Isosceles triangle shown above has two sides with length  and one side length . The length of side   inches. Side length . Find the perimeter of the triangle. 

Possible Answers:

 inches

 inches

 inches

 inches

 inches

Correct answer:

 inches

Explanation:

To find the perimeter of this triangle, apply the perimeter formula:



Since, , and  then  must have a value of: 

This triangle has two side lengths of  inches, and one side length of  inches. 

Thus, the solution is:

Example Question #12 : Isosceles Triangles

A triangle has two sides with length  and one side length . The length of side   yard. If the length of   the length of side , what is the perimeter of the triangle? 

Possible Answers:

 yard

 yard

 yard

 yard

 yard

Correct answer:

 yard

Explanation:

The first step to solving this problem is that we must find the length of length  Since,  is 4 the length of side , use the following steps: 




Now, apply the formula: 




Example Question #1 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

An isosceles triangle has an angle of 110°.  Which of the following angles could also be in the triangle?

Possible Answers:

35

90

110

20

55

Correct answer:

35

Explanation:

An isosceles triangle always has two equal angles. As there cannot be another 110° (the triangle cannot have over 180° total), the other two angles must equal eachother. 180° - 110° = 70°. 70° represents the other two angles, so it needs to be divided in 2 to get the answer of 35°. 

Example Question #2 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

An isosceles triangle ABC is laid flat on its base.  Given that <B, located in the lower left corner, is 84 degrees, what is the measurement of the top angle, <A?

Possible Answers:
96
42
84
12
20
Correct answer: 12
Explanation:

Since the triangle is isosceles, and <A is located at the top of the triangle that is on its base, <B and <C are equivalent.  Since <B is 84 degrees, <C is also.  There are 180 degrees in a triangle so 180 - 84 - 84 = 12 degrees.

Example Question #1 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

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Triangle ABC is isosceles

x and y are positive integers

 

A

---

x

 

B

---

y

Possible Answers:

The two quantities are equal

Quantity B is greater

Quantity A is greater

The relationship cannot be determined

Correct answer:

Quantity B is greater

Explanation:

Since we are given expressions for the two congruent angles of the isosceles triangle, we can set the expressions equal to see how x relates to y. We get,

x – 3 = y – 7 --> y = x + 4

Logically, y must be the greater number if it takes x an additional 4 units to reach its value (knowing they are both positive integers).

Example Question #4 : How To Find An Angle In An Acute / Obtuse Isosceles Triangle

An isosceles triangle has one obtuse angle that is . What is the value of one of the other angles?

Possible Answers:

Correct answer:

Explanation:

We know that an isosceles triangel has two equal sides and thus two equal angles opposite those equal sides. Because there is one obtuse angle of 112 degrees we automatically know that this angle is the vertex. If you sum any triangle's interior angles, you always get 180 degrees. 

180 – 112 = 68 degrees. Thus there are 68 degrees left for the two equal angles. Each angle must therefore measure 34 degrees.

 Isosceles-triangle-1

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