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SSAT Upper Level Math : SSAT Upper Level Quantitative (Math)

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

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All SSAT Upper Level Math Resources

6 Diagnostic Tests 311 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #581 : Geometry

$\bigtriangleup ABC \sim \bigtriangleup DEF$ ; AB = AC = 50, BC = 40 .

Which of the following is true about $\bigtriangleup DEF$ ?

Possible Answers:

$\bigtriangleup DEF$ is isosceles and obtuse.

$\bigtriangleup DEF$ is scalene and obtuse.

$\bigtriangleup DEF$ is isosceles and acute.

$\bigtriangleup DEF$ is scalene and acute.

None of the other responses is correct.

Correct answer:

$\bigtriangleup DEF$ is isosceles and acute.

Explanation:

$\bigtriangleup ABC \sim \bigtriangleup DEF$ , so corresponding sides are in proportion; it follows that

$\frac{DE}{AB} = \frac{DF}{AC}$

$\frac{DE}{50} = \frac{DF}{50 }$

DE = DF

Therefore, $\bigtriangleup DEF$ is isosceles.

Also, corresponding angles are congruent, so if $\bigtriangleup ABC$ acute (or obtuse), so is $\bigtriangleup DEF$ . We can compare the sum of the squares of the lesser two sides to that of the greatest;

$40^{2}+50^{2}\; \square \; 50^{2}$

$1,600+2,500\; \square \; 2,500$

$4,100\; \square \; 2,500$

$4,100\; > \; 2,500$

The sum of the squares of the lesser sides is greater than the square of the greatest side, so $\bigtriangleup ABC$ is acute - and so is $\bigtriangleup DEF$ . The correct response is that $\bigtriangleup DEF$ is isosceles and acute.

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Example Question #142 : Properties Of Triangles

Which of the following statements would prove that the statement

$\bigtriangleup ABC \sim \bigtriangleup DEF$

is false?

Possible Answers:

$m \angle A + m \angle B \ne m \angle D + m \angle E$

$AB + BC \ne DE+EF$

$\bigtriangleup ABC$ and $\bigtriangleup DEF$ have different perimeters

None of the other statements alone would prove the statement $\bigtriangleup ABC \sim \bigtriangleup DEF$ to be false.

$\bigtriangleup ABC$ and $\bigtriangleup DEF$ have different areas

Correct answer:

$m \angle A + m \angle B \ne m \angle D + m \angle E$

Explanation:

Triangles that are similar need not have congruent sides, so it does not follow that AB + BC = DE+EF , or that their perimeters are equal. Consequently, their areas need not be equal either.

However, if $\bigtriangleup ABC \sim \bigtriangleup DEF$ , then corresponding angles are congruent; specifically, $m \angle A = m \angle D$ and $m \angle B =m \angle E$ . Therefore, $m \angle A + m \angle B = m \angle D + m \angle E$ . Contrapositively, if $m \angle A + m \angle B \ne m \angle D + m \angle E$ , then $\bigtriangleup ABC \nsim \bigtriangleup DEF$ .

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Example Question #15 : How To Find If Two Acute / Obtuse Triangles Are Similar

$\bigtriangleup ABC \sim \bigtriangleup DEF \sim \bigtriangleup GHI$ .

AB = 30, DE = 40, DF = 60, EF = 50, GI = 150 .

What is the ratio of the area of $\bigtriangleup GHI$ to that of $\bigtriangleup ABC$ ?

Possible Answers:

36:1

100:9

16:1

225:2

625:36

Correct answer:

100:9

Explanation:

The similarity ratio of two triangles is the ratio of the lengths of their corresponding sides.

The similarity ratio of $\bigtriangleup DEF$ to $\bigtriangleup ABC$ is

$\frac{DE}{AB} = \frac{40}{30} = \frac{4}{3}$ .

The similarity ratio of $\bigtriangleup GHI$ to $\bigtriangleup DEF$ is

$\frac{GI}{DF} = \frac{150}{60} = \frac{5}{2}$

Multipliy these to get the similarity ratio of $\bigtriangleup GHI$ to $\bigtriangleup ABC$ :

$\frac{4}{3} \times \frac{5}{2} = \frac{20}{6}= \frac{10}{3}$

The ratio of the areas of two similar figures is the square of their similarity ratio, so the ratio of the areas of the triangles is

$\left ( \frac{10}{3} \right )^{2} = \frac{100}{9}$ .

The correct choice is 100:9 .

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Example Question #591 : Geometry

Given: $\bigtriangleup ABC$ and $\bigtriangleup DEF$ ; $\frac{AB}{DE} = \frac{AC}{DF}$ .

Which of the following statements would not be enough, along with what is given, to prove that $\bigtriangleup ABC \sim \bigtriangleup DEF$ ?

Possible Answers:

$\angle A \cong \angle D$

$\angle C \cong \angle F$

$\frac{AC}{DF}= \frac{BC}{EF}$

The given information is enough to prove the triangles similar.

$\frac{AB}{DE} = \frac{BC}{EF}$

Correct answer:

$\angle C \cong \angle F$

Explanation:

From both the given proportion statement and either $\frac{AB}{DE} = \frac{BC}{EF}$ or $\frac{AC}{DF}= \frac{BC}{EF}$ , it follows that $\frac{AB}{DE} =\frac{AC}{DF}= \frac{BC}{EF}$ —all three pairs of corresponding sides are in proportion; by the Side-Side-Side Similarity Theorem, $\bigtriangleup ABC \sim \bigtriangleup DEF$ . From the given proportion statement and $\angle A \cong \angle D$ , since these are the included angles of the sides that are in proportion, then by the Side-Angle-Side Similarity Theorem, $\bigtriangleup ABC \sim \bigtriangleup DEF$ . From the given proportion statement and $\angle C \cong \angle F$ , since these are nonincluded angles of the sides that are in proportion, no similarity can be deduced.

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Example Question #1 : How To Find The Height Of An Acute / Obtuse Triangle

The area of a triangle is 240 , and the base of the triangle is . What is the height for this triangle?

Possible Answers:

Correct answer:

Explanation:

Use the formula to find the area of a triangle.

$\text{Area}=\frac{base\times height}{2}$

Now, plug in the values for the area and the base to solve for height .

$240=\frac{30 \times h}{2}$

480=30h

h=16

The height of the triangle is .

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Example Question #1 : How To Find The Height Of An Acute / Obtuse Triangle

A triangle has an area of 360m^2 and a base of 60m . In meters, find the height.

Possible Answers:

12m

24m

18m

Correct answer:

12m

Explanation:

Use the formula to find the area of a triangle.

$\text{Area}=\frac{base\times height}{2}$

Now, plug in the values for the area and the base to solve for height .

$360=\frac{60\times h}{2}$

720=60h

h=12

The height of the triangle is meters.

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Example Question #61 : Acute / Obtuse Triangles

A triangle has an area of and a base of . Find the height.

Possible Answers:

Correct answer:

Explanation:

Use the formula to find the area of a triangle.

$\text{Area}=\frac{base\times height}{2}$

Now, plug in the values for the area and the base to solve for height .

$45=\frac{15\times h}{2}$

90=15h

h=6

The height of the triangle is .

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Example Question #1 : How To Find The Height Of An Acute / Obtuse Triangle

A triangle has an area of and a base of . Find the height.

Possible Answers:

Correct answer:

Explanation:

Use the formula to find the area of a triangle.

$\text{Area}=\frac{base\times height}{2}$

Now, plug in the values for the area and the base to solve for height .

$24=\frac{12\times h}{2}$

48=12h

h=4

The height of the triangle is .

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Example Question #153 : Properties Of Triangles

A triangle has an area of and a base of . Find the height.

Possible Answers:

Correct answer:

Explanation:

Use the formula to find the area of a triangle.

$\text{Area}=\frac{base\times height}{2}$

Now, plug in the values for the area and the base to solve for height .

$54=\frac{12\times h}{2}$

108=12h

h=9

The height of the triangle is .

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Example Question #154 : Properties Of Triangles

A triangle has an area of and a base of . Find the height.

Possible Answers:

Correct answer:

Explanation:

Use the formula to find the area of a triangle.

$\text{Area}=\frac{base\times height}{2}$

Now, plug in the values for the area and the base to solve for height .

$60=\frac{12\times h}{2}$

120=12h

h=10

The height of the triangle is .

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SSAT Upper Level Math : SSAT Upper Level Quantitative (Math)

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

All SSAT Upper Level Math Resources

Example Questions

Example Question #581 : Geometry

Example Question #142 : Properties Of Triangles

Example Question #15 : How To Find If Two Acute / Obtuse Triangles Are Similar

Example Question #591 : Geometry

Example Question #1 : How To Find The Height Of An Acute / Obtuse Triangle

Example Question #1 : How To Find The Height Of An Acute / Obtuse Triangle

Example Question #61 : Acute / Obtuse Triangles

Example Question #1 : How To Find The Height Of An Acute / Obtuse Triangle

Example Question #153 : Properties Of Triangles

Example Question #154 : Properties Of Triangles

All SSAT Upper Level Math Resources

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