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SAT Math : How to find the length of the side of an acute / obtuse triangle

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Example Question #21 : Acute / Obtuse Triangles

Given $\bigtriangleup ABC$ with $m \angle A > m \angle B$ and $m \angle C = 45^{\circ }$ .

Which of the following could be the correct ordering of the lengths of the sides of the triangle?

I) AB > BC > AC

II) BC > AB > AC

III) BC > AC > AB

Possible Answers:

II only

II or III only

I only

I or II only

III only

Correct answer:

II or III only

Explanation:

Given two angles of unequal measure in a triangle, the side opposite the greater angle is longer than the side opposite the other angle.

If $\angle C$ were the angle of greatest measure, then $m \angle A , m \angle B < 45^{\circ }$ , and

$m \angle A + m \angle B + m \angle C < 45 ^{\circ}+ 45 ^{\circ}+ 45 ^{\circ} = 135 ^{\circ} < 180 ^{\circ}$ .

Since the measures of the angles must total $180 ^{\circ}$ , $\angle C$ cannot have the greatest measure.

$m \angle A > m \angle B$ , so we can explore two other possibilities:

$m \angle A > m \angle B > m \angle C$ , which here is possible if, for example,

$m \angle A = 85 ^{\circ }$ and $m \angle B = 50 ^{\circ }$ - since 85 + 50 + 45 = 180 ; and,

$m \angle A > m \angle C > m \angle B$ , which here is possible if, for example,

$m \angle A = 95 ^{\circ }$ and $m \angle B = 40 ^{\circ }$ - since 95 + 40 + 45 = 180 .

If $m \angle A > m \angle B > m \angle C$ , then BC > AC > AB ; if $m \angle A > m \angle C > m \angle B$ , then BC > AB > AC .

This makes the correct response II or III only.

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Example Question #152 : Geometry

Which of the given answers choices could NOT represent the sides of a triangle?

Possible Answers:

4, 5, 10

2, 2, 3

5, 6, 10

10, 10, 19

All of the given answers could represent the sides of a triangle.

Correct answer:

4, 5, 10

Explanation:

In order for three lengths to represent the sides of a triangle, they must pass the Triangle Inequality.

a + b >c

This means that the sum of any two sides of the triangle must exceed the length of the third side.

With the given answers, the one set of lengths that fail this test is 4, 5, 10 .

$4+5 \ngtr 10$

Therefore, the lengths 4, 5, 10 could not represent the sides of a triangle.

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SAT Math : How to find the length of the side of an acute / obtuse triangle

Study concepts, example questions & explanations for SAT Math

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Example Question #21 : Acute / Obtuse Triangles

Example Question #152 : Geometry

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