ISEE Upper Level Quantitative : How to find the length of the hypotenuse of a right triangle : Pythagorean Theorem

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

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Example Questions

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Example Question #1 : Right Triangles

What is the hypotenuse of a right triangle with sides 5 and 8?

Possible Answers:

√89

15

5√4

12

Correct answer:

√89

Explanation:

Because this is a right triangle, we can use the Pythagorean Theorem which says a2 + b2 = c2, or the squares of the two sides of a right triangle must equal the square of the hypotenuse. Here we have a = 5 and b = 8.

a2 + b2 = c2

52 + 82 = c2

25 + 64 = c2

89 = c2

c = √89

Example Question #2 : Right Triangles

Which is the greater quantity?

(a) The hypotenuse of a  right triangle with a leg of length 20

(b) The hypotenuse of a right triangle with legs of length 19 and 21

Possible Answers:

It is impossible to tell from the information given

(b) is greater

(a) and (b) are equal

(a) is greater

Correct answer:

(b) is greater

Explanation:

The hypotenuses of the triangles measure as follows:

(a) 

(b) 

, so , making (b) the greater quantity

Example Question #3 : Right Triangles

Which is the greater quantity?

(a) The hypotenuse of a right triangle with legs  and .

(b) The hypotenuse of a right triangle with legs  and .

Possible Answers:

(a) is greater.

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

Correct answer:

(a) is greater.

Explanation:

The hypotenuses of the triangles measure as follows:

(a) 

(b) 

, so , making (a) the greater quantity.

Example Question #4 : Right Triangles

A right triangle has a leg  feet long and a hypotenuse  feet long. Which is the greater quantity?

(a) The length of the second leg of the triangle

(b) 60 inches

Possible Answers:

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

(a) is greater.

Correct answer:

(a) is greater.

Explanation:

The length of the second leg can be calculated using the Pythagorean Theorem. Set :

The second leg therefore measures  inches.

 

Example Question #5 : Right Triangles

What is the hypotenuse of a right triangle with sides 9 inches and 12 inches?

Possible Answers:

Correct answer:

Explanation:

Since we're dealing with right triangles, we can use the Pythagorean Theorem (). In this formula, a and b are the sides, while c is the hypotenuse. The hypotenuse of a right triangle is the longest side and the side that is opposite the right angle. Now, we can plug into our formula, which looks like this: We simplify and get . At this point, isolate c. This means taking the square root of both sides so that your answer is 15in.

Example Question #1 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Right_triangle

The perimeter of a regular pentagon is 75% of that of the triangle in the above diagram. Which is the greater quantity?

(A) The length of one side of the pentagon

(B) One and one-half feet

Possible Answers:

(A) and (B) are equal

(A) is greater

(B) is greater

It is impossible to determine which is greater from the information given

Correct answer:

(B) is greater

Explanation:

By the Pythagorean Theorem, the hypotenuse of the right triangle is 

 inches, making its perimeter

 inches.

The pentagon in question has sides of length 75% of 112, or 

.

Since a pentagon has five sides of equal length, each side will have measure

 inches.

One and a half feet are equivalent to  inches, so (B) is the greater quantity.

Example Question #7 : Right Triangles

Right_triangle

The track at Gauss High School is unusual in that it is shaped like a right triangle, as shown above. 

Cary decides to get some exercise by running from point A to point B, then running half of the distance from point B to point C. 

Which is the greater quantity?

(A) The distance Cary runs

(B) One-fourth of a mile

Possible Answers:

(B) is greater

(A) is greater

(A) and (B) are equal

It is impossible to determine which is greater from the information given

 

Correct answer:

(B) is greater

Explanation:

By the Pythagorean Theorem, the distance from B to C is 

  feet

Cary runs 

 feet

 

Since 5,280 feet make a mile, one-fourth of a mile is equal to 

 feet.

(B) is greater

 

Example Question #8 : Right Triangles

Right_triangle

Give the length of the hypotenuse of the above right triangle in terms of .

Possible Answers:

Correct answer:

Explanation:

If we let  be the length of the hypotenuse, then by the Pythagorean theorem,

Example Question #9 : Right Triangles

In Square  is the midpoint of  is the midpoint of , and  is the midpoint of . Construct the line segments  and .

Which is the greater quantity?

(a) 

(b) 

Possible Answers:

(b) is the greater quantity

It cannot be determined which of (a) and (b) is greater

(a) is the greater quantity

(a) and (b) are equal

Correct answer:

(b) is the greater quantity

Explanation:

The figure referenced is below:
Square x

For the sake of simplicity, assume that the square has sides of length 4. The following reasoning is independent of the actual lengths, and the reason for choosing 4 will become apparent in the explanation.

 and  are midpoints of their respective sides, so , making  the hypotenuse of a triangle with legs of length 2 and 2. Therefore,

.

Also, , and since  is the midpoint of . , making  the hypotenuse of a triangle with legs of length 1 and 4. Therefore, 

, so 

Example Question #10 : Right Triangles

Untitled

Figure NOT drawn to scale.

In the above figure,  is a right angle. 

What is the length of  ? 

Possible Answers:

Correct answer:

Explanation:

The altitude of a right triangle from the vertex of its right angle divides the triangle into two smaller triangles each similar to the larger triangle. In particular, 

.

Their corresponding sides are in proportion, so, setting the ratios of the hypotenuses to the short legs equal to each other,

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