How to find the length of the side of a triangle - ISEE Upper Level (grades 9-12) Quantitative Reasoning

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Question

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

Answer

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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Question

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

Which is the greater quantity?

(a)

(b)

Answer

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

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

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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Question

$\Delta ABC$ is acute; . Which is the greater quantity?

(a)

(b)

Answer

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

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

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

$AC< \sqrt{169}$

(b) is the greater quantity.

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Question

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

(a)

(b)

Answer

To compare the lengths of $\overline{AB}$ and $\overline{BC}$ from the angle measures, it is necessary to know which of their opposite angles - $\angleC$ $\angle C$ and $\angle A$ , 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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Question

$\Delta ABC$ has obtuse angle $\angle B$ ; . Which is the greater quantity?

(a)

(b)

Answer

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

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

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

$AC > \sqrt{625}$

,

so (a) is the greater quantity.

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Question

Given $\Delta ABC$ with . Which is the greater quantity?

(a)

(b)

Answer

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 ; if $\angle B$ is obtuse, then .

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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Question

Given $\Delta ABC$ with . Which is the greater quantity?

(a)

(b)

Answer

Use the Triangle Inequality:

This makes (b) the greater quantity.

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Question

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

(a) 18

(b)

Answer

Suppose there exists a second triangle $\bigtriangleup DEF$ such that and . 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

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

and

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

Between $\bigtriangleup ABC$ and $\bigtriangleup DEF$ , 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,

.

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Question

Two sides of a triangle have length 8 inches and 6 inches. Which of the following lengths of the third side would make the triangle isosceles?

Answer

An isosceles triangle, by definition, has two sides of equal length. Having the third side measure either 6 inches or 8 inches would make the triangle meet this criterion. Also, since 6 inches and 8 inches are equal to $\frac{1}{2} \textrm{ ft}$ and $\frac{2}{3} \textrm{ ft}$ , respectively, these also make the triangle isosceles. Therefore, the correct choice is that all four make the triangle isosceles.

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Question

$\Delta ABC$ is an isosceles triangle with obtuse angle $\angle A$ .

Which is the greater quantity?

(a)

(b)

Answer

A triangle must have at least two acute angles; if $\angle A$ is obtuse, then $\angle B$ and $\angle C$ are the acute angles of $\Delta ABC$ . Since $\Delta ABC$ is isosceles, the Isosceles Triangle Theorem requires two of the angles to be congruent; they must be the two acute angles $\angle B$ and $\angle C$ . Also, the sides opposite these two angles are the congruent sides; these sides are $\overline{AB}$ and $\overline{AC}$ , respectively. This makes the quantities (a) and (b) equal.

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Question

Note: Figure NOT drawn to scale.

Refer to the above diagram. Which expression is equivalent to ?

Answer

This is an isosceles triangle, so the left and right sides are of equal length. Draw the altitude of this triangle, as follows:

The altitude is a perpendicular bisector of the base; it is one leg of a right triangle with half the base, which is 15 inches, as the other leg, and one side, which is inches, as the hypotenuse. By definition,

$\cos 40 ^{\circ } = \frac{15}{x}$ (adjacent side divided by hypotenuse), so

$\frac{x}{\cos 40 ^{\circ }} \cdot \cos 40 ^{\circ } =\frac{x}{\cos 40 ^{\circ }} \cdot \frac{15}{x}$

$x= \frac{15}{\cos 40 ^{\circ }}$

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