ISEE Upper Level Math : Right Triangles

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

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Example Questions

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Example Question #2 : 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 #2 : Triangles

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 #2 : 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 #4 : How To Find The Length Of The Side Of A Right Triangle

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 #5 : How To Find The Length Of The Side Of A Right Triangle

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

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 #1 : 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,

 

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

 is a right angle; .

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(a) is greater.

(b) is greater.

It is impossible to tell from the information given.

(a) and (b) are equal.

Correct answer:

(a) and (b) are equal.

Explanation:

. Corresponding angles of similar triangles are congruent, so since  is a right angle, so is 

The hypotenuse  of  is twice as long as leg ; by the  Theorem, . Again, by similiarity, 

.

Example Question #5 : Apply The Pythagorean Theorem To Determine Unknown Side Lengths In Right Triangles: Ccss.Math.Content.8.G.B.7

If a right triangle has a base of   and a height of , what is the length of the hypotenuse?

Possible Answers:

Correct answer:

Explanation:

To solve this problem, we must utilize the Pythagorean Theorom, which states that:

We know that the base is , so we can substitute in for .  We also know that the height is , so we can substitute in for .


Next we evaluate the exponents:

Now we add them together:

Then, .

is not a perfect square, so we simply write the square root as  .

Example Question #6 : Apply The Pythagorean Theorem To Determine Unknown Side Lengths In Right Triangles: Ccss.Math.Content.8.G.B.7

If a right triangle has a base of and a height of , what is the length of the hypotenuse?

Possible Answers:

Correct answer:

Explanation:

To solve this problem, we are going to use the Pythagorean Theorom, which states that .

We know that this particular right triangle has a base of , which can be substituted for , and a height of , which can be substituted for . If we rewrite the theorom using these numbers, we get:

Next, we evaluate the expoenents:

Then, .

To solve for , we must find the square root of . Since this is not a perfect square, our answer is simply .

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