SSAT Upper Level Math : Geometry

Study concepts, example questions & explanations for SSAT Upper Level Math

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Example Questions

Example Question #9 : Acute / Obtuse Triangles

Find the perimeter of a triangle with side lengths \(\displaystyle 45, 40,\text{ and }35\).

Possible Answers:

\(\displaystyle 130\)

\(\displaystyle 120\)

\(\displaystyle 115\)

\(\displaystyle 110\)

Correct answer:

\(\displaystyle 120\)

Explanation:

To find the perimeter of a triangle, add up all of its sides.

\(\displaystyle \text{Perimeter}=45+35+40=120\)

Example Question #10 : Acute / Obtuse Triangles

Bill has a triangular garden that he needs to fence. The garden has side lengths of \(\displaystyle \text{9 feet, 10 feet, and 12 feet}\). In feet, how much fencing will Bill need?

Possible Answers:

\(\displaystyle 36ft\)

\(\displaystyle 35ft\)

\(\displaystyle 31ft\)

\(\displaystyle 22ft\)

Correct answer:

\(\displaystyle 31ft\)

Explanation:

To find how much fencing Bill needs, you will need to find the perimeter of the triangle. The perimeter of a triangle is found by adding up all the sides together.

\(\displaystyle \text{Perimeter}=9+10+12=31\)

Example Question #541 : Geometry

Find the perimeter of a triangle with side lengths of \(\displaystyle 12, 12, 18\).

Possible Answers:

\(\displaystyle 32\)

\(\displaystyle 22\)

\(\displaystyle 52\)

\(\displaystyle 42\)

Correct answer:

\(\displaystyle 42\)

Explanation:

To find the perimeter of a triangle, add up all of its sides.

\(\displaystyle \text{Perimeter}=12+12+18=42\)

Example Question #542 : Geometry

Find the perimeter of the triangle below:

4

Possible Answers:

\(\displaystyle 14\)

\(\displaystyle 12\)

\(\displaystyle 13\)

\(\displaystyle 11\)

Correct answer:

\(\displaystyle 13\)

Explanation:

First, find the length of the missing side.

Since the angle measurements in the base of the triangle are the same, this is an isosceles triangle. The side lengths directly across from the equal angles must be the same. Thus, the missing side length is \(\displaystyle 5\).

Now, to find the perimeter, add up all the side lengths.

\(\displaystyle \text{Perimeter}=5+5+3=13\)

Example Question #543 : Geometry

Find the perimeter of the triangle below:

3

Possible Answers:

\(\displaystyle 50\)

\(\displaystyle 40\)

\(\displaystyle 60\)

\(\displaystyle 70\)

Correct answer:

\(\displaystyle 50\)

Explanation:

First, find the length of the missing side.

Since the angle measurements in the base of the triangle are the same, this is an isosceles triangle. The side lengths directly across from the equal angles must be the same. Thus, the missing side length is \(\displaystyle 16\).

Now, to find the perimeter, add up all the side lengths.

\(\displaystyle \text{Perimeter}=16+16+18=50\)

Example Question #544 : Geometry

Find the perimeter of the triangle below:

2

Possible Answers:

\(\displaystyle 61\)

\(\displaystyle 57\)

\(\displaystyle 47\)

\(\displaystyle 60\)

Correct answer:

\(\displaystyle 57\)

Explanation:

First, find the length of the missing side.

Since there are two angles that are the same, this is an isosceles triangle. The side lengths directly across from the equal angles must be the same. Thus, the missing side length is \(\displaystyle 18\).

Now, to find the perimeter, add up all the side lengths.

\(\displaystyle \text{Perimeter}=18+18+21=57\)

Example Question #545 : Geometry

Find the perimeter of the triangle below:

1

Possible Answers:

\(\displaystyle 48\)

\(\displaystyle 58\)

\(\displaystyle 44\)

\(\displaystyle 54\)

Correct answer:

\(\displaystyle 48\)

Explanation:

First, find the length of the missing side.

Since there are two angles that are the same, this is an isosceles triangle. The side lengths directly across from the equal angles must be the same. Thus, the missing side length is \(\displaystyle 14\).

Now, to find the perimeter, add up all the side lengths.

\(\displaystyle \text{Perimeter}=14+14+20=48\)

Example Question #1 : How To Find An Angle In An Acute / Obtuse Triangle

Triangle_a

Figure NOT drawn to scale.

If \(\displaystyle x = 72^{\circ }\) and \(\displaystyle y = 43^{\circ }\), evaluate \(\displaystyle w\).

Possible Answers:

\(\displaystyle 115^{\circ }\)

\(\displaystyle 137^{\circ}\)

\(\displaystyle 108^{\circ}\)

\(\displaystyle 125^{\circ }\)

\(\displaystyle 65^{\circ}\)

Correct answer:

\(\displaystyle 115^{\circ }\)

Explanation:

The measure of an exterior angle of a triangle is the sum of the measures of its remote interior angles, so 

\(\displaystyle w = x + y = 72 + 43 = 115 ^{\circ }\)

Example Question #2 : How To Find An Angle In An Acute / Obtuse Triangle

If the vertex angle of an isoceles triangle is \(\displaystyle 64^{\circ}\), what is the value of one of its base angles?

Possible Answers:

\(\displaystyle 26^{\circ}\)

\(\displaystyle 116^{\circ}\)

\(\displaystyle 36^{\circ}\)

\(\displaystyle 64^{\circ}\)

\(\displaystyle 58^{\circ}\)

Correct answer:

\(\displaystyle 58^{\circ}\)

Explanation:

In an isosceles triangle, the base angles are the same. Also, the three angles of a triangle add up to \(\displaystyle 180^{\circ}\).

So, subtract the vertex angle from \(\displaystyle 180^{\circ}\). You get \(\displaystyle 116^{\circ}\).

Because there are two base angles you divide \(\displaystyle 116^{\circ}\) by \(\displaystyle 2\), and you get \(\displaystyle 58^{\circ}\).

Example Question #2 : How To Find An Angle In An Acute / Obtuse Triangle

Triangle

Note: Figure NOT drawn to scale.

Refer to the above diagram. 

\(\displaystyle 45^{\circ } \leq m \angle 1 \leq 50^{\circ }\)

\(\displaystyle 60^{\circ } \leq m \angle 2 \leq 70 ^{\circ }\)

Which of the following could be a measure of \(\displaystyle \angle 3\) ?

Possible Answers:

\(\displaystyle 100^{\circ }\)

\(\displaystyle 110^{\circ }\)

\(\displaystyle 125^{\circ }\)

All of the other choices give a possible measure of \(\displaystyle \angle 3\).

\(\displaystyle 130^{\circ }\)

Correct answer:

\(\displaystyle 110^{\circ }\)

Explanation:

The measure of an exterior angle of a triangle is the sum of the measures of its remote interior angles, so 

\(\displaystyle \angle 3 = \angle 1 + \angle 2\).

We also have the following constraints:

\(\displaystyle 45^{\circ } \leq m \angle 1 \leq 50^{\circ }\)

\(\displaystyle 60^{\circ } \leq m \angle 2 \leq 70 ^{\circ }\)

Then, by the addition property of inequalities,

\(\displaystyle 45^{\circ } + 60 ^{\circ }\leq m \angle 1 +m \angle 2 \leq 50^{\circ } + 70 ^{\circ }\)

\(\displaystyle 105 ^{\circ }\leq m \angle 3 \leq 120^{\circ }\)

Therefore, the measure of \(\displaystyle \angle 3\) must fall in that range. Of the given choices, only \(\displaystyle 110^{\circ }\) falls in that range.

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