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

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Example Questions

Example Question #1 : Right Triangles

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

Possible Answers:

√89

5√4

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

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Example Question #2 : Right Triangles

Which is the greater quantity?

(a) The hypotenuse of a $45^{\circ}-45^{\circ}-90^{\circ}$ right triangle with a leg of length 20

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

Possible Answers:

(a) and (b) are equal

It is impossible to tell from the information given

(a) is greater

(b) is greater

Correct answer:

(b) is greater

Explanation:

The hypotenuses of the triangles measure as follows:

(a) $c = \sqrt {20^{2} + 20^{2} } = \sqrt {400 + 400 } = \sqrt {800}$

(b) $c = \sqrt {19^{2} + 21^{2} } = \sqrt {361 + 441} = \sqrt {802}$

800 < 802 , so $\sqrt {800} < \sqrt {802}$ , making (b) the greater quantity

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Example Question #3 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

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:

(b) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

(a) is greater.

Correct answer:

(a) is greater.

Explanation:

The hypotenuses of the triangles measure as follows:

(a) $c = \sqrt {10^{2} + 15^{2} } = \sqrt {100 + 225 } = \sqrt {325}$

(b) $c = \sqrt {12^{2} + 13^{2} } = \sqrt {144 + 169 } = \sqrt {313}$

325 > 313 , so $\sqrt {325} > \sqrt {313}$ , making (a) the greater quantity.

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Example Question #4 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

A right triangle has a leg $4 \frac{1}{2}$ feet long and a hypotenuse $7 \frac{1}{2}$ 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.

(a) is greater.

(a) and (b) are equal.

(b) is greater.

Correct answer:

(a) is greater.

Explanation:

The length of the second leg can be calculated using the Pythagorean Theorem. Set $c = 7 \frac{1}{2} = \frac{15}{2} , a =4 \frac{1}{2} = \frac{9}{2}$ :

$b ^{2} = c ^{2}-a ^{2}$

$b ^{2} = \left ( \frac{15}{2} \right ) ^{2}- \left ( \frac{9}{2} \right )^{2}\textup{}$

$b ^{2} = \frac{225}{4} -\frac{81}{4}$

$b ^{2} = \frac{144}{4}$

$b ^{2} = 36$

$b = \sqrt{36} = 6$

The second leg therefore measures $6 \times 12 = 72$ inches.

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Example Question #5 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

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

Possible Answers:

25in

10.5in

27in

225in

15 in

Correct answer:

15 in

Explanation:

Since we're dealing with right triangles, we can use the Pythagorean Theorem ( a^2+b^2=c^2 ). 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: 9^2+12^2=c^2. We simplify and get 225=c^2 . At this point, isolate c. This means taking the square root of both sides so that your answer is 15in.

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Example Question #6 : 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) is greater

(B) is greater

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

(A) and (B) are equal

Correct answer:

(B) is greater

Explanation:

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

$\sqrt{14^{2}+48^{2}} = \sqrt{196+2,304} = \sqrt{2,500} = 50$ inches, making its perimeter

14 + 48 + 50 =112 inches.

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

$112 \times 0.75 = 84$ .

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

$84 \div 5 = 16 \frac{4}{5}$ inches.

One and a half feet are equivalent to $12 \times 1 \frac{1}{2} = 18$ inches, so (B) is the greater quantity.

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Example Question #7 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

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:

(A) and (B) are equal

(B) is greater

(A) 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 distance from B to C is

$\sqrt{600^{2} + 800^{2} }$

$= \sqrt{360,000 + 640,000 }$

$= \sqrt{1,000,000} = 1,000$ feet

Cary runs

$800 + \frac{1}{2} \times1,000 = 800 + 500 = 1,300$ feet

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

$5,280 \div 4 = 1,320$ feet.

(B) is greater

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Example Question #8 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Right_triangle

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

Possible Answers:

$2k ^{2} +8k$

$\sqrt{2k ^{2} +8}$

$\sqrt{2k ^{2} +8k}$

$2 \sqrt{2k}$

$2k ^{2} +8$

Correct answer:

$\sqrt{2k ^{2} +8}$

Explanation:

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

$c = \sqrt{(k+2)^{2}+(k-2)^{2}}$

$c = \sqrt{(k ^{2}+4k+4) +(k ^{2}-4k+4)}$

$c = \sqrt{2k ^{2} +8}$

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

In Square SARE . is the midpoint of $\overline{SE}$ , is the midpoint of $\overline{SA}$ , and is the midpoint of $\overline{QA}$ . Construct the line segments $\overline{QX}$ and $\overline{UR}$ .

Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

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

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 QS = SX = 2 , making $\overline{QX}$ the hypotenuse of a triangle with legs of length 2 and 2. Therefore,

$(QX ) ^{2}= (SQ)^{2}+ (SX)^{2} = 2^{2}+ 2^{2} = 4+ 4 = 8$ .

Also, QA = 2 , and since is the midpoint of $\overline{AQ}$ , AU = 1 . AR = 4 , making $\overline{UR}$ the hypotenuse of a triangle with legs of length 1 and 4. Therefore,

$(UR)^{2} = (AU)^{2}+ (AR)^{2} = 1^{2}+ 4^{2} = 1+ 16 = 17$

$(UR)^{2} >(QX ) ^{2}$ , so UR > QX

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Example Question #3 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Untitled

Figure NOT drawn to scale.

In the above figure, $\angle ABC$ is a right angle.

What is the length of $\overline{AC}$ ?

Possible Answers:

$33 \frac{4}{5 }$

$31\frac{1}{5}$

$38 \frac{4}{5 }$

$36\frac{1}{5}$

Correct answer:

$33 \frac{4}{5 }$

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,

$\bigtriangleup BXC \sim \bigtriangleup ABC$ .

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

$\frac{AC}{BC} = \frac{BC}{CX}$

$\frac{AC}{13} = \frac{13}{5}$

$\frac{AC}{13} \cdot 13 = \frac{13}{5} \cdot 13$

$AC = \frac{169}{5} = 33 \frac{4}{5 }$

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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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #1 : Right Triangles

Example Question #2 : Right Triangles

Example Question #3 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #4 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #5 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #6 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #7 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #8 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

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

Example Question #3 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

All ISEE Upper Level Quantitative Resources

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