SSAT Upper Level Quantitative - Properties of Triangles

Find the angle value of .

$46^{\circ}$

$56^{\circ}$

$36^{\circ}$

$66^{\circ}$

Explanation

All the angles in a triangle add up to degrees.

$90^{\circ}+44^{\circ}+y=180^{\circ}$

$134^{\circ}+y=180^{\circ}$

$134^{\circ}+y-134^{\circ}=180^{\circ}-134^{\circ}$

$y=46^{\circ}$

The area of a right triangle is $50\textup{ in}^2$ . If the base of the triangle is $10\textup{ in}$ , what is the length of the height, in inches?

Explanation

To find the height, plug what is given in the question into the formula used to find the area of a triangle.

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

Use the information given in the question:

$\text{Base}=10$

$\text{Area}=50$

$50=\frac{10\times height}{2}$

Now, solve for the height.

$10\times height=100$

A given right triangle has two legs of lengths and , respectively. What is the area of the triangle?

Not enough information to solve

Explanation

The area of a right triangle with a base and a height can be found with the formula $A=\frac{1}{2}bh$ . Since the two legs of a right triangle are perpendicular to each other, we can use these as the base and height in the formula. Therefore:

$A=\frac{1}{2}bh$

$A=\frac{1}{2}(6:cm)(8:cm)$

$A=\frac{1}{2}(48:cm^2)$

The lengths of the hypotenuses of ten similar right triangles form an arithmetic sequence. The smallest triangle has legs of lengths 3 and 4 inches; the second-smallest triangle has a hypotenuse of length one foot.

Which of the following responses comes closest to the area of the largest triangle?

8 square feet

7 square feet

6 square feet

9 square feet

5 square feet

Explanation

The hypotenuse of the smallest triangle can be calculated using the Pythagorean Theorem:

$c = \sqrt{a^{2}+b^{2}} = \sqrt{3^{2}+4^{2}} = \sqrt{9+16} = \sqrt{25}=5$ inches.

Let $c_{n}$ be the lengths of the hypotenuses of the triangles in inches. $c_{1} = 5$ and $c_{2} = 12$ , so their common difference is

$d = c_{2} - c_{1} = 12 - 5= 7$

The arithmetic sequence formula is

$c_{n} = c_{1} + (n-1)d$

The length of the hypotenuse of the largest triangle - the tenth triangle - can be found by substituting $c_{1} = 5, n= 10, d = 7$ :

$c_{10} =5+ (10-1) 7 = 5 + 9 \cdot 7 = 5 + 63 = 68$ inches.

The largest triangle has hypotenuse of length 68 inches. Since the triangles are similar, corresponding sides are in proportion. If we let and be the lengths of the legs of the largest triangle, then

$\frac{a}{3} = \frac{68}{5}$

$\frac{a}{3} \cdot 3 = \frac{68}{5} \cdot 3$

Similarly,

$\frac{b}{4} = \frac{68}{5}$

$\frac{b}{4} \cdot 4 = \frac{68}{5} \cdot 4$

The area of a right triangle is half the product of its legs:

$A = \frac{ab}{2}= \frac{40.8 \cdot 54.4}{2} = 1,109.76$ square inches.

Divide this by 144 to convert to square feet:

$1,109.76 \div 144 \approx 7.7$

Of the given responses, 8 square feet is the closest, and is the correct choice.

A right triangle has a hypotenuse of and one leg has a length of . What is the length of the other leg?

$3\sqrt{7}$

$7\sqrt{3}$

Explanation

When calculating the lengths of sides of a right triangle, we can use the Pythagorean Theorem as follows:

$a^{2}+b^{2}=c^{2}$ , where and are legs of the triangle and is the hypotenuse.

Plugging in our given values:

$9^{2}+b^{2}=12^{2}$

Subtracting from each side of the equation:

$b^{2}=12^{2}-9^{2}$

$b^{2}=144-81$

$b^{2}=63$

Taking the square root of each side of the equation:

$b=\sqrt{63}$

Simplifying the square root:

$b=\sqrt{9\times 7}$

$b=3\sqrt{7}$

The perimeter of an equilateral triangle is 39 meters. In meters, how long is one side of the triangle?

Explanation

An equilateral triangle has sides that are the same length. Then, to find the length of one side, you only need to divide the perimeter by 3.

$39\div3=13$

Find the angle value of .

$46^{\circ}$

$56^{\circ}$

$36^{\circ}$

$66^{\circ}$

Explanation

All the angles in a triangle add up to degrees.

$90^{\circ}+44^{\circ}+y=180^{\circ}$

$134^{\circ}+y=180^{\circ}$

$134^{\circ}+y-134^{\circ}=180^{\circ}-134^{\circ}$

$y=46^{\circ}$

$\bigtriangleup ABC \sim \bigtriangleup DEF$ ; ; $\bigtriangleup DEF$ has perimeter 300.

Evaluate .

Insufficient information is given to answer the problem.

Explanation

The ratio of the perimeters of two similar triangles is equal to the ratio of the lengths of a pair of corresponding sides. Therefore,

$\frac{DE}{AB}= \frac{P_{\bigtriangleup DEF}}{P_{\bigtriangleup ABC }}$ and $\frac{EF}{BC}= \frac{P_{\bigtriangleup DEF}}{P_{\bigtriangleup ABC }}$ , or

$\frac{DE}{AB}=\frac{EF}{BC} = \frac{P_{\bigtriangleup DEF}}{P_{\bigtriangleup ABC }}$

By one of the properties of proportions, it follows that

$\frac{DE+EF}{AB+BC} = \frac{P_{\bigtriangleup DEF}}{P_{\bigtriangleup ABC }}$

The perimeter of $\bigtriangleup ABC$ is

, so

$\frac{DE+EF}{24+20} = \frac{300}{80}$

$\frac{DE+EF}{44} = \frac{300}{80}$

$\frac{DE+EF}{44} \cdot 44 = \frac{300}{80} \cdot 44$

Which of the following responses comes closest to the area of the largest triangle?

8 square feet

7 square feet

6 square feet

9 square feet

5 square feet

Explanation

The hypotenuse of the smallest triangle can be calculated using the Pythagorean Theorem:

$c = \sqrt{a^{2}+b^{2}} = \sqrt{3^{2}+4^{2}} = \sqrt{9+16} = \sqrt{25}=5$ inches.

Let $c_{n}$ be the lengths of the hypotenuses of the triangles in inches. $c_{1} = 5$ and $c_{2} = 12$ , so their common difference is

$d = c_{2} - c_{1} = 12 - 5= 7$

The arithmetic sequence formula is

$c_{n} = c_{1} + (n-1)d$

The length of the hypotenuse of the largest triangle - the tenth triangle - can be found by substituting $c_{1} = 5, n= 10, d = 7$ :

$c_{10} =5+ (10-1) 7 = 5 + 9 \cdot 7 = 5 + 63 = 68$ inches.

The largest triangle has hypotenuse of length 68 inches. Since the triangles are similar, corresponding sides are in proportion. If we let and be the lengths of the legs of the largest triangle, then

$\frac{a}{3} = \frac{68}{5}$

$\frac{a}{3} \cdot 3 = \frac{68}{5} \cdot 3$

Similarly,

$\frac{b}{4} = \frac{68}{5}$

$\frac{b}{4} \cdot 4 = \frac{68}{5} \cdot 4$

The area of a right triangle is half the product of its legs:

$A = \frac{ab}{2}= \frac{40.8 \cdot 54.4}{2} = 1,109.76$ square inches.

Divide this by 144 to convert to square feet:

$1,109.76 \div 144 \approx 7.7$

Of the given responses, 8 square feet is the closest, and is the correct choice.

A right triangle has a hypotenuse of and one leg has a length of . What is the length of the other leg?

$3\sqrt{7}$

$7\sqrt{3}$

Explanation

When calculating the lengths of sides of a right triangle, we can use the Pythagorean Theorem as follows:

$a^{2}+b^{2}=c^{2}$ , where and are legs of the triangle and is the hypotenuse.

Plugging in our given values:

$9^{2}+b^{2}=12^{2}$

Subtracting from each side of the equation:

$b^{2}=12^{2}-9^{2}$

$b^{2}=144-81$

$b^{2}=63$

Taking the square root of each side of the equation:

$b=\sqrt{63}$

Simplifying the square root:

$b=\sqrt{9\times 7}$

$b=3\sqrt{7}$

Properties of Triangles

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