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ISEE Upper Level Quantitative : Right Triangles

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All ISEE Upper Level Quantitative Resources

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

Right triangle 5

Figure NOT drawn to scale.

Refer to the above triangle. Which is the greater quantity?

(a)

(b) 108

Possible Answers:

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

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

Correct answer:

(b) is the greater quantity

Explanation:

We can compare these numbers by comparing their squares.

By the Pythagorean Theorem,

$(AC) ^{2}= (AB) ^{2}+ (BC)^{2} = 100 ^{2}+ 40 ^{2} = 10,000 + 1,600 = 11,600$

Also,

$108^{2} = 11,664$

$108^{2} >(AC) ^{2}$ , so 108 > AC .

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

Consider a triangle, $\bigtriangleup ABC$ , in which AB = 33 , BC = 44 , and $m \angle B = 100 ^{\circ}$ . Which is the greater quantity?

(a) 55

(b)

Possible Answers:

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

(a) is the greater quantity

(a) and (b) are equal

(b) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

Suppose AC = 55 .

By the Converse of the Pythagorean Theorem, a triangle is right if and only if the sum of the squares of the lengths of the smallest two sides is equal to the square of the longest side. Compare the quantities $(AB)^{2}+ ( BC )^{2}$ and $(AC)^{2}$

$(AB)^{2}+ ( BC )^{2} = 33^{2} + 44 ^{2} = 1,089+ 1,936= 3,025$

$(AC)^{2} = 55^{2} = 3,025$

Therefore, if AC = 55

$(AB)^{2}+ ( BC )^{2} =(AC)^{2}$ , so $\bigtriangleup ABC$ is right, with the right angle opposite longest side $\overline{AC}$ . Thus, $\angle B$ is right and has degree measure 90.

However, $\angle B$ has degree measure greater than 90, so, as a consequence of the Converse of the Pythagorean Theorem and the SAS Inequality Theorem, it holds that AC > 55 .

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Example Question #51 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

$\Delta ABC$ and $\Delta DEF$ are right triangles, with right angles $\angle B , \angle E$ , respectively.

AC = DF = 10

Which is the greater quantity?

(a) The perimeter of $\Delta ABC$

(b) The perimeter of $\Delta DEF$

Possible Answers:

It is impossible to tell from the information given.

(b) is greater.

(a) and (b) are equal.

(a) is greater.

Correct answer:

It is impossible to tell from the information given.

Explanation:

No information is given about the legs of either triangle; therefore, no information about their perimeters can be deduced.

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Example Question #52 : Plane Geometry

Right_triangle

Note: Figure NOT drawn to scale

Refer to the above triangle. Starting at point A, an insect walks clockwise along the sides of the triangle until he has walked 75% of the length of $\overline{CB}$ . What percent of the perimeter of the triangle has the insect walked?

Possible Answers:

$73 \frac{1}{3} \%$

$83 \frac{1}{3} \%$

$72 \%$

$78 \%$

$60 \%$

Correct answer:

$73 \frac{1}{3} \%$

Explanation:

By the Pythagorean Theorem, the distance from B to C, which we will call , is equal to

$D = \sqrt{13^{2}- 5^{2}} = \sqrt{169- 25 }= \sqrt{144} = 12$ .

The perimeter of the triangle is

5+12 + 13 = 30 .

The insect traveled the entirety of the hypotenuse, which is 13 units long, and 75% of the longer leg, which adds 75% of 12, or $0.75 \times 12 = 9$ units. Therefore, the insect has traveled 22 out of the 30 units perimeter, or

$\frac{22}{30} \times 100 = 73 \frac{1}{3} \%$ of the perimeter.

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Example Question #53 : Plane Geometry

Right triangle

Refer to the above diagram, in which $\bigtriangleup ABC$ is a right triangle with altitude $\overline{BX}$ . Which is the greater quantity?

(a) Four times the perimeter of $\bigtriangleup AXB$

(b) Three times the perimeter of $\bigtriangleup CXB$

Possible Answers:

(b) is the greater quantity

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

(a) is the greater quantity

(a) and (b) are equal

Correct answer:

(a) and (b) are equal

Explanation:

The altitude of a right triangle from the vertex of its right angle - which, here, is $\overline{BX}$ - divides the triangle into two triangles similar to each other. The ratio of the hypotenuse of $\bigtriangleup CXB$ to that of $\bigtriangleup AXB$ (which are corresponding sides) is

$\frac{BC}{AB} = \frac{40}{30} =\frac{4}{3}$ ,

making this the similarity ratio. The ratio of the perimeters of two similar triangles is the same as their similarity ratio; therefore, if is the perimeter of $\bigtriangleup AXB$ and is the perimeter of $\bigtriangleup CXB$ , it follows that

$\frac{Y}{X} = \frac{4}{3}$

$\frac{Y}{X} \cdot X = \frac{4}{3} \cdot X$

$Y = \frac{4}{3} X$

Multiply both sides by 3:

$3 \cdot Y =3 \cdot \frac{4}{3} X$

3Y = 4X

Three times the perimeter of $\bigtriangleup CXB$ is therefore equal to four times that of $\bigtriangleup AXB$ .

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Example Question #54 : Plane Geometry

Quantity A: The hypotenuse of a right triangle with sides and .

Quantity B: The height of a triangle with an area of and base of .

Possible Answers:

Quantity A is greater.

The two quantities are equal.

Quantity B is greater.

The relationship of the quantities cannot be determined.

Correct answer:

The two quantities are equal.

Explanation:

Quantity A: This is the special -- triangle, where the two sides have lengths of and and the hypotenuse is .

Quantity B: This triangle has an area of and base of . The area of a triangle is $\frac{bh}{2}$ , so that height must be .

Quantity A and Quantity B are equal.

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Example Question #52 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Right_triangle

Note: Figure NOT drawn to scale.

Refer to the above figure.

Which is the greater quantity?

(a)

(b) $30 ^{\circ }$

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:

Since the shorter leg of the right triangle is half the hypotenuse, the triangle is a $30 ^{\circ } -60^{\circ } -90^{\circ }$ triangle, with the $30 ^{\circ }$ angle opposite the shorter leg. That makes x = 30 .

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Example Question #56 : Plane Geometry

Right triangle $\Delta ABC$ has right angle $\angle B$ .

$m\angle A = \left ( x + 30 \right )^ {\circ} ,m\angle C = \left ( 2x- 57\right ) ^ {\circ}$

Which is the greater quantity?

(a) $m\angle A$

(b) $m\angle C$

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) is greater

Explanation:

The degree measures of the acute angles of a right triangle total 90, so we solve for in the following equation:

$\left ( x + 30 \right )+ \left ( 2x- 57\right ) = 90$

3x-27 = 90

3x-27+ 27 = 90 + 27

3x = 117

$3x \div 3 = 117 \div 3$

x= 39

(a) $m\angle A = \left ( x + 30 \right )^ {\circ} = \left ( 39 + 30 \right )^ {\circ} = 69^ {\circ}$

(b) $m\angle C =\left ( 90 - 69 \right ) ^ {\circ} = 21^ {\circ}$

$m\angle A > m\angle C$

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Example Question #21 : Right Triangles

$\bigtriangleup ABC \sim \bigtriangleup DEF$

$\angle A$ is a right angle.

Which is the greater quantity?

(a) $m \angle D$

(b) $m \angle E$

Possible Answers:

(a) and (b) are equal

(a) is the greater quantity

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

(b) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

Corresponding angles of similar triangles are congruent, so, since $\bigtriangleup ABC \sim \bigtriangleup DEF$ , and $\angle A$ is right, it follows that

$m \angle D = m \angle A = 90 ^{\circ }$

$\angle D$ is a right angle of a right triangle $\bigtriangleup DEF$ . The other two angles must be acute - that is, with measure less than $90 ^{\circ }$ - so $m \angle D > m \angle E$ .

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Example Question #22 : Right Triangles

$\bigtriangleup ABC$ is inscribed in a circle. $\angle B$ is a right angle, and $m \overarc{ACB} = 270 ^{\circ }$ .

Which is the greater quantity?

(a) $m \angle A$

(b) $m \angle C$

Possible Answers:

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

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

Correct answer:

(a) and (b) are equal

Explanation:

The figure referenced is below:

Inscribed angle

$\overarc{ACB}$ has measure $270 ^{\circ }$ , so its corresponding minor arc, $\overarc{A B}$ , has measure $360 ^{\circ } - 270 ^{\circ } = 90^{\circ }$ . The inscribed angle that intercepts this arc, which is $\angle C$ , has measure half this, or $45 ^{\circ }$ . Since $\angle B$ is a right angle, the other acute angle, $\angle A$ , has measure

$m \angle A = 90 ^{\circ } - m \angle C = 90 ^{\circ } - 45^{\circ } = 45 ^{\circ }$

Therefore, $m \angle A = 45 ^{\circ } = m \angle C$ .

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : Right Triangles

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

All ISEE Upper Level Quantitative Resources

Example Questions

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

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

Example Question #51 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Example Question #52 : Plane Geometry

Example Question #53 : Plane Geometry

Example Question #54 : Plane Geometry

Example Question #52 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Example Question #56 : Plane Geometry

Example Question #21 : Right Triangles

Example Question #22 : Right Triangles

All ISEE Upper Level Quantitative Resources

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