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SAT Math : Acute / Obtuse Triangles

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Example Question #6 : How To Find The Length Of The Side Of An Acute / Obtuse Triangle

If 2 sides of the triangle are have lengths equal to 8 and 14, what is one possible length of the third side?

Possible Answers:

Not enough information

Correct answer:

Explanation:

The sum of the lengths of 2 sides of a triangle must be greater than—but not equal to—the length of the third side. Further, the third side must be longer than the difference between the greater and the lesser of the other two sides; therefore, 20 is the only possible answer.

$4+8< 14$

$6+8= 14$

$8+14> 20$

$8+14=22$

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Example Question #561 : Geometry

In $\Delta ABC$ the length of AB is 15 and the length of side AC is 5. What is the least possible integer length of side BC?

Possible Answers:

Correct answer:

Explanation:

Rule - the length of one side of a triangle must be greater than the differnce and less than the sum of the lengths of the other two sides.

Given lengths of two of the sides of the $\Delta ABC$ are 15 and 5. The length of the third side must be greater than 15-5 or 10 and less than 15+5 or 20.

The question asks what is the least possible integer length of BC, which would be 11

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Example Question #41 : Acute / Obtuse Triangles

Two sides of a triangle have lengths 4 and 7. Which of the following represents the set of all possible lengths of the third side, x?

Possible Answers:

3 < x < 7

2 < x < 12

4 < x < 11

3 < x < 11

4 < x < 7

Correct answer:

3 < x < 11

Explanation:

The set of possible lengths is: 7-4 < x < 7+4, or 3 < X < 11.

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Example Question #561 : Geometry

If two sides of a triangle have lengths 8 and 10, what could the length of the third side NOT be?

Possible Answers:

Correct answer:

Explanation:

According to the Triangle Inequality Theorem, the sums of the lengths of any two sides of a triangle must be greater than the length of the third side. Since 10 + 8 is 18, the only length out of the answer choices that is not possible is 19.

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Example Question #10 : How To Find The Length Of The Side Of An Acute / Obtuse Triangle

Given $\bigtriangleup ABC$ with $m \angle A > m \angle B$ and $m \angle C = 60^{\circ }$ .

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:

I only.

II only.

II or III only.

I or II only.

III only.

Correct answer:

II 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. Therefore, we seek to find the relationship among the measures of the angles.

$m \angle C = 60^{\circ }$ , and the measures of the interior angles of a triangle total $180^{\circ }$ , so

$m \angle A + m \angle B + m \angle C = 180 ^{\circ }$

$m \angle A + m \angle B + 60 ^{\circ } = 180 ^{\circ }$

$m \angle A + m \angle B = 120 ^{\circ }$

Since $m \angle A > m \angle B$ ,

$m \angle A + m \angle A > m \angle B+ m \angle A = 120^{\circ }$

$2 m \angle A > 120^{\circ }$

$m \angle A > 60^{\circ }$ ,

and, similarly,

$m \angle B < 60^{\circ }$

Therefore,

$m \angle A > m \angle C > m \angle B$ ,

and the lengths of their opposite sides rank similarly:

BC > AB > AC .

The correct response is that only (II) can be true.

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Example Question #11 : How To Find The Length Of The Side Of An Acute / Obtuse Triangle

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 or III only

III only

II only

I or II only

I 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 : Sat Mathematics

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

Possible Answers:

4, 5, 10

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

10, 10, 19

5, 6, 10

2, 2, 3

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 : Acute / Obtuse Triangles

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

Example Questions

Example Question #6 : How To Find The Length Of The Side Of An Acute / Obtuse Triangle

Example Question #561 : Geometry

Example Question #41 : Acute / Obtuse Triangles

Example Question #561 : Geometry

Example Question #10 : How To Find The Length Of The Side Of An Acute / Obtuse Triangle

Example Question #11 : How To Find The Length Of The Side Of An Acute / Obtuse Triangle

Example Question #152 : Sat Mathematics

All SAT Math Resources

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