ISEE Upper Level Math : Triangles

Study concepts, example questions & explanations for ISEE Upper Level Math

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Example Questions

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Example Question #1 : Isosceles Triangles

Two sides of an isosceles triangle have lengths 3 feet and 4 feet. Which of the following could be the length of the third side?

Possible Answers:

Correct answer:

Explanation:

An isosceles triangle, by definition, has two sides of equal length. Having the third side measure either 3 feet or 4 feet would make the triangle meet this criterion.

3 feet is equal to  inches, and 4 feet is equal to  inches. We choose 36 inches, since that, but not 48 inches, is a choice.

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

The triangles are similar. Solve for .

Question_12

Possible Answers:

Correct answer:

Explanation:

Because the triangles are similar, proportions can be used to solve for the length of the side:

Cross-multiply:

Example Question #3 : Isosceles Triangles

One of the base angles of an isosceles triangle is . Give the measure of the vertex angle.

Possible Answers:

Correct answer:

Explanation:

The base angles of an isosceles triangle are always equal. Therefore both base angles are .

Let the measure of the third angle. Since the sum of the angles of a triangle is , we can solve accordingly:

Example Question #1 : Right Triangles

A right triangle has a hypotenuse of 10 and a side of 6. What is the missing side?

Possible Answers:

Correct answer:

Explanation:

To find the missing side, use the Pythagorean Theorem . Plug in (remember c is always the hypotenuse!) so that . Simplify and you get Subtract 36 from both sides so that you get Take the square root of both sides. B is 8.

Example Question #41 : Plane Geometry

Right_triangle

Refer to the above diagram. Which of the following quadratic equations would yield the value of  as a solution?

Possible Answers:

Correct answer:

Explanation:

By the Pythagorean Theorem,

Example Question #3 : Right Triangles

Right_triangle

Note: Figure NOT drawn to scale.

Refer to the above diagram. Which of the following quadratic equations would yield the value of  as a solution?

Possible Answers:

Correct answer:

Explanation:

By the Pythagorean Theorem,

Example Question #1 : Right Triangles

Right_triangle

Note: Figure NOT drawn to scale.

Refer to the above diagram.

Find the length of .

Possible Answers:

Correct answer:

Explanation:

First, find .

Since  is an altitude of right  to its hypotenuse, 

 by the Angle-Angle Postulate, so 

Example Question #2 : Triangles

Right_triangle

Note: Figure NOT drawn to scale.

Refer to the above diagram.

Find the length of .

Possible Answers:

Correct answer:

Explanation:

First, find .

Since  is an altitude of  from its right angle to its hypotenuse, 

 by the Angle-Angle Postulate, so 

Example Question #6 : Right Triangles

Right_triangle

Note: Figure NOT drawn to scale.

Refer to the above diagram. Evaluate .

Possible Answers:

Correct answer:

Explanation:

By the Pythagorean Theorem,

 

Example Question #7 : Right Triangles

A right triangle  with hypotenuse  is inscribed in , a circle with radius 26. If , evaluate the length of .

Possible Answers:

Insufficient information is given to answer the question.

Correct answer:

Explanation:

The arcs intercepted by a right angle are both semicircles, so hypotenuse  shares its endpoints with two semicircles. This makes  a diameter of the circle, and .

By the Pythagorean Theorem,

 

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