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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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All ISEE Upper Level Quantitative Resources

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Example Questions

Example Question #11 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Untitled

Figure NOT drawn to scale.

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

What is the length of $\overline{AB}$ $\displaystyle \overline{AB}$ ?

Possible Answers:

$31\frac{1}{5}$ $\displaystyle 31\frac{1}{5}$

$36\frac{1}{5}$ $\displaystyle 36\frac{1}{5}$

$33 \frac{4}{5 }$ $\displaystyle 33 \frac{4}{5 }$

$38 \frac{4}{5 }$ $\displaystyle 38 \frac{4}{5 }$

Correct answer:

$31\frac{1}{5}$ $\displaystyle 31\frac{1}{5}$

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,

$\bigtriangleup BXC \sim \bigtriangleup ABC$ $\displaystyle \bigtriangleup BXC \sim \bigtriangleup ABC$.

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

$\frac{AB}{BC} = \frac{BX}{XC}$ $\displaystyle \frac{AB}{BC} = \frac{BX}{XC}$

By the Pythagorean Theorem.

$(BX)^{2} = (BC)^{2} - (CX)^{2}$ $\displaystyle (BX)^{2} = (BC)^{2} - (CX)^{2}$

$(BX)^{2} = 13^{2} - 5^{2} = 169 - 25 = 144$ $\displaystyle (BX)^{2} = 13^{2} - 5^{2} = 169 - 25 = 144$

$BX = \sqrt{144} = 12$ $\displaystyle BX = \sqrt{144} = 12$

The proportion statement becomes

$\frac{AB}{13} = \frac{12}{5}$ $\displaystyle \frac{AB}{13} = \frac{12}{5}$

$\frac{AB}{13} \cdot 13 = \frac{12}{5} \cdot 13$ $\displaystyle \frac{AB}{13} \cdot 13 = \frac{12}{5} \cdot 13$

$AB = \frac{156}{5}= 31 \frac{1}{5}$ $\displaystyle AB = \frac{156}{5}= 31 \frac{1}{5}$

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Example Question #41 : Plane Geometry

Given: $\bigtriangleup ABC$ $\displaystyle \bigtriangleup ABC$ with AB = 6 $\displaystyle AB = 6$, BC= 8 $\displaystyle BC= 8$, AC = 12 $\displaystyle AC = 12$.

Which is the greater quantity?

(a) $m \angle B$ $\displaystyle m \angle B$

(b) $90^{\circ }$ $\displaystyle 90^{\circ }$

Possible Answers:

(a) and (b) are equal

(b) is the greater quantity

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

(a) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

The measure of the angle formed by the two shorter sides of a triangle can be determined to be acute, right, or obtuse by comparing the sum of the squares of those lengths to the square of the length of the opposite side. We compare:

$(AB) ^{2} + (BC)^{2} = 6^{2} + 8 ^{2} = 36 + 64 = 100$ $\displaystyle (AB) ^{2} + (BC)^{2} = 6^{2} + 8 ^{2} = 36 + 64 = 100$

$(AC)^{2} = 12^{2} = 144$ $\displaystyle (AC)^{2} = 12^{2} = 144$

$(AB) ^{2} + (BC)^{2} < (AC) ^{2}$ $\displaystyle (AB) ^{2} + (BC)^{2} < (AC) ^{2}$; it follows that $\angle B$ $\displaystyle \angle B$ is obtuse, and has measure greater than $90 ^{\circ }$ $\displaystyle 90 ^{\circ }$

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Example Question #41 : Triangles

Untitled

Figure NOT drawn to scale.

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

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

Possible Answers:

$36\frac{1}{5}$ $\displaystyle 36\frac{1}{5}$

$38 \frac{4}{5 }$ $\displaystyle 38 \frac{4}{5 }$

$31\frac{1}{5}$ $\displaystyle 31\frac{1}{5}$

$36\frac{1}{3}$ $\displaystyle 36\frac{1}{3}$

$33 \frac{4}{5 }$ $\displaystyle 33 \frac{4}{5 }$

Correct answer:

$33 \frac{4}{5 }$ $\displaystyle 33 \frac{4}{5 }$

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,

$\bigtriangleup BXC \sim \bigtriangleup ABC$ $\displaystyle \bigtriangleup BXC \sim \bigtriangleup ABC$.

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

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

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

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

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

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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 #11 : How To Find The Length Of The Hypotenuse Of A Right Triangle : Pythagorean Theorem

Example Question #41 : Plane Geometry

Example Question #41 : Triangles

All ISEE Upper Level Quantitative Resources

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