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ISEE Upper Level Quantitative : How to find the length of the side of a triangle

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All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #11 :Triangles

Which of the following could be the lengths of the three sides of a scalene triangle?

Possible Answers:

$8 \textrm{ cm, } 9 \textrm{ cm, }10 \textrm{ cm}$

$6 \textrm{ in, } 8 \textrm{ in, }10 \textrm{ in}$

$5,000 \textrm{ m, } 4,000 \textrm{ m, }8,000 \textrm{ m}$

All of the other choices are possible lengths of a scalene triangle

$0.7 \textrm{ in, } 0.9 \textrm{ in, }1.4 \textrm{ in}$

Correct answer:

All of the other choices are possible lengths of a scalene triangle

Explanation:

A scalene triangle, by definition, has sides all of different lengths. Since all of the given choices fit that criterion, the correct choice is that all can be scalene.

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

鉴于 $\Delta ABC$ with right angle $\angle B$ , $m \angle C = 50 ^{\circ }$ .

Which is the greater quantity?

(a)

(b)

Possible Answers:

It is impossible to tell from the information given

(a) and (b) are equal

(b) is greater

(a) is greater

Correct answer:

(a) is greater

Explanation:

$m \angle B = 90^{\circ}$

$m \angle C = 50 ^{\circ }$

The sum of the measures of the angles of a triangle is 180, so

$m \angle A + m \angle B + m \angle C = 180$

$m \angle A +90 + 50 = 180$

$m \angle A +140= 180$

$m \angle A +140-140= 180-140$

$m \angle A = 40^{\circ }$

$m \angle C > m \angle A$ , so the side opposite $\angle C$ , which is $\overline{AB}$ , is longer than the side opposite $\angle A$ , which is $\overline{BC}$ . This makes (a) the greater quantity.

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Example Question #3 :How To Find The Length Of The Side Of A Triangle

鉴于 $\Delta ABC$ with obtuse angle $\angle A$ , which is the greater quantity?

(a)

(b)

Possible Answers:

(b) is greater.

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

Correct answer:

(b) is greater.

Explanation:

To compare the lengths of $\overline{AB}$ 一个d $\overline{BC}$ from the angle measures, it is necessary to know which of their opposite angles - $\angleC$ $\angle C$ 一个d $\angle A$ , respectively - is the greater angle. Since $\angle A$ is the obtuse angle, it has the greater measure, and $\overline{BC}$ is the longer side. This makes (b) greater.

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Example Question #4 :How To Find The Length Of The Side Of A Triangle

$\Delta ABC$ has obtuse angle $\angle B$ ; AB = 7, BC = 24 . Which is the greater quantity?

(a)

(b)

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:

Since $\angle B$ is the obtuse angle of $\Delta ABC$ ,

$\left ( AC \right )^{2} >\left ( AB \right )^{2} + \left ( BC \right )^{2} = 7^{2} + 24^{2} = 49 + 576 = 625$ .

$\left ( AC \right )^{2} > 625$ ,

$AC > \sqrt{625}$

AC >25 ,

so (a) is the greater quantity.

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Example Question #5 :How To Find The Length Of The Side Of A Triangle

鉴于 $\Delta ABC$ with AB = 5, BC = 12 . Which is the greater quantity?

(a)

(b)

Possible Answers:

(a) is greater.

(a) and (b) are equal.

(b) is greater.

It is impossible to tell from the information given.

Correct answer:

(b) is greater.

Explanation:

Use the Triangle Inequality:

AC < AB + BC

AC < 5 + 12

AC < 17

This makes (b) the greater quantity.

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Example Question #6 :How To Find The Length Of The Side Of A Triangle

鉴于 $\Delta ABC$ with AB = 5, BC = 12 . Which is the greater quantity?

(a)

(b)

Possible Answers:

(b) is greater.

(a) is greater.

(a) and (b) are equal.

It is impossible to tell from the information given.

Correct answer:

It is impossible to tell from the information given.

Explanation:

By the Converse of the Pythagorean Theorem,

$AC = \sqrt{(AB)^{2}+(BC)^{2}}= \sqrt{5^{2}+12^{2}}= \sqrt{25+144}=\sqrt{169} = 13$

if and only if $\angle B$ is a right angle.

However, if $\angle B$ is acute, then AC < 13 ; if $\angle B$ is obtuse, then AC > 13 .

Since we do not know whether $\angle B$ is acute, right, or obtuse, we cannot determine whether (a) or (b) is greater.

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Example Question #7 :How To Find The Length Of The Side Of A Triangle

$\Delta ABC$ is acute; AB = 5, BC = 12 . Which is the greater quantity?

(a)

(b)

Possible Answers:

It is impossible to tell from the information given.

(a) and (b) are equal.

(a) is greater.

(b) is greater.

Correct answer:

(b) is greater.

Explanation:

Since $\Delta ABC$ is an acute triangle, $\angle B$ is an acute angle, and

$\left ( AC \right )^{2} < \left ( AB \right )^{2} + \left ( BC \right )^{2} = 5^{2} + 12^{2} = 25 + 144 = 169$

$\left ( AC \right )^{2} < 169$ ,

$AC< \sqrt{169}$

AC < 13

(b) is the greater quantity.

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Example Question #21 :Geometry

鉴于: $\bigtriangleup ABC$ . $AB= AC = 12, m \angle A = 80 ^{\circ }$ . Which is the greater quantity?

(a) 18

(b)

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) is the greater quantity

Explanation:

Suppose there exists a second triangle $\bigtriangleup DEF$ such that DE = DF = 12 一个d EF = 18 . Whether $\angle D$ , the angle opposite the longest side, is acute, right, or obtuse can be determined by comparing the sum of the squares of the lengths of the shortest sides to the square of the length of the longest:

$(DE)^{2}+ (DF)^{2} = 12^{2}+ 12^{2} = 144 + 144 = 288$

$(EF) ^{2}= 18 ^{2} = 324$

$(EF) ^{2}> (DE)^{2}+ (DF)^{2}$ , making $\angle D$ obtuse, so $m \angle D > 90 ^{\circ }$ .

We know that

$AB= AC = 12, m \angle A = 80 ^{\circ }$

一个d

$DE = DF = 12, m \angle D > m \angle A$ .

Between $\bigtriangleup ABC$ 一个d $\bigtriangleup DEF$ , we have two sets of congruent sides, with the included angle of the latter of greater measure than that of the former. It follows from the Side-Angle-Side Inequality (or Hinge) Theorem that between the third sides, $\overline{EF}$ is the longer. Therefore,

BC < EF = 18 .

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All ISEE Upper Level Quantitative Resources

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ISEE Upper Level Quantitative : How to find the length of the side of a triangle

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #11 :Triangles

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

Example Question #3 :How To Find The Length Of The Side Of A Triangle

Example Question #4 :How To Find The Length Of The Side Of A Triangle

Example Question #5 :How To Find The Length Of The Side Of A Triangle

Example Question #6 :How To Find The Length Of The Side Of A Triangle

Example Question #7 :How To Find The Length Of The Side Of A Triangle

Example Question #21 :Geometry

All ISEE Upper Level Quantitative Resources

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