ISEE Upper Level Quantitative : Right Triangles

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

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Example Questions

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

Right triangle 5

Figure NOT drawn to scale.

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

(a) \displaystyle AC

(b) 108

Possible Answers:

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

(a) and (b) are equal

(a) is the greater quantity

(b) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

We can compare these numbers by comparing their squares.

By the Pythagorean Theorem, 

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

Also,

\displaystyle 108^{2} = 11,664

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

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

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

(a) 55

(b) \displaystyle AC

Possible Answers:

(a) is the greater quantity

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

(a) and (b) are equal

(b) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

Suppose \displaystyle 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 \displaystyle (AB)^{2}+ ( BC )^{2} and \displaystyle (AC)^{2}

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

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

Therefore, if \displaystyle AC = 55

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

However, \displaystyle \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 \displaystyle AC > 55.

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

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

\displaystyle AC = DF = 10

Which is the greater quantity?

(a) The perimeter of \displaystyle \Delta ABC

(b) The perimeter of \displaystyle \Delta DEF

Possible Answers:

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

(a) and (b) are equal.

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.

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 \displaystyle \overline{CB}. What percent of the perimeter of the triangle has the insect walked?

Possible Answers:

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

\displaystyle 60 \%

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

\displaystyle 78 \%

\displaystyle 72 \%

Correct answer:

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

Explanation:

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

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

The perimeter of the triangle is 

\displaystyle 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 \displaystyle 0.75 \times 12 = 9 units. Therefore, the insect has traveled 22 out of the 30 units perimeter, or 

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

Example Question #53 : Plane Geometry

Right triangle

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

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

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

Possible Answers:

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

(a) and (b) are equal

(b) is the greater quantity

(a) is the greater quantity

Correct answer:

(a) and (b) are equal

Explanation:

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

\displaystyle \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 \displaystyle X is the perimeter of \displaystyle \bigtriangleup AXB and \displaystyle Y is the perimeter of \displaystyle \bigtriangleup CXB, it follows that

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

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

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

Multiply both sides by 3:

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

\displaystyle 3Y = 4X

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

Example Question #53 : Geometry

Quantity A: The hypotenuse of a right triangle with sides \displaystyle 3 and \displaystyle 4.

Quantity B: The height of a triangle with an area of \displaystyle 10 and base of \displaystyle 4

Possible Answers:

Quantity B is greater. 

The relationship of the quantities cannot be determined. 

The two quantities are equal.

Quantity A is greater. 

Correct answer:

The two quantities are equal.

Explanation:

Quantity A:  This is the special \displaystyle 3-\displaystyle 4-\displaystyle 5 triangle, where the two sides have lengths of \displaystyle 3 and \displaystyle 4 and the hypotenuse is \displaystyle 5.

Quantity B: This triangle has an area of \displaystyle 10 and base of \displaystyle 4. The area of a triangle is \displaystyle \frac{bh}{2}, so that height must be \displaystyle 5

Quantity A and Quantity B are equal.

Example Question #53 : 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) \displaystyle x

(b) \displaystyle 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 \displaystyle 30 ^{\circ } -60^{\circ } -90^{\circ } triangle, with the \displaystyle 30 ^{\circ } angle opposite the shorter leg. That makes \displaystyle x = 30.

Example Question #53 : Triangles

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

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

Which is the greater quantity?

(a) \displaystyle m\angle A

(b) \displaystyle m\angle C

Possible Answers:

(a) and (b) are equal

(a) is greater

(b) is greater

It is impossible to tell from the information given

Correct answer:

(a) is greater

Explanation:

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

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

\displaystyle 3x-27 = 90

\displaystyle 3x-27+ 27 = 90 + 27

\displaystyle 3x = 117

\displaystyle 3x \div 3 = 117 \div 3

\displaystyle x= 39

 

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

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

 

\displaystyle m\angle A > m\angle C

Example Question #61 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

\displaystyle \bigtriangleup ABC \sim \bigtriangleup DEF 

\displaystyle \angle A is a right angle.

Which is the greater quantity?

(a) \displaystyle m \angle D

(b) \displaystyle m \angle E

Possible Answers:

(a) is the greater quantity

(a) and (b) are equal

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 \displaystyle \bigtriangleup ABC \sim \bigtriangleup DEF, and \displaystyle \angle A is right, it follows that 

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

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

Example Question #61 : Geometry

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

Which is the greater quantity? 

(a) \displaystyle m \angle A

(b) \displaystyle m \angle C

Possible Answers:

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

(a) is the greater quantity

(b) is the greater quantity

(a) and (b) are equal

Correct answer:

(a) and (b) are equal

Explanation:

The figure referenced is below:

Inscribed angle

 

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

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

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

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