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SSAT Middle Level Math : Plane Geometry

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Example Question #41 : Geometry

The hypotenuse of a right triangle is 25 inches; it has one leg 15 inches long. Give its area in square feet.

Possible Answers:

$1 \frac{1}{24} \textrm{ ft}^{2}$

$1 \frac{1}{12} \textrm{ ft}^{2}$

$2 \frac{29}{48} \textrm{ ft}^{2}$

$1 \frac{13}{24}\textrm{ ft} ^{2}$

$2 \frac{5}{48} \textrm{ ft}^{2}$

Correct answer:

$1 \frac{1}{24} \textrm{ ft}^{2}$

Explanation:

The area of a right triangle is half the product of the lengths of its legs, so we need to use the Pythagorean Theorem to find the length of the other leg. Set c = 25, b = 15 :

$a^{2} = c^{2} - b^{2}$

$a^{2} = 25^{2} - 15^{2}$

$a^{2} = 625-225$

$a^{2} =400$

$a = \sqrt{400} = 20$

The legs are 15 and 20 inches long. Divide both dimensions by 12 to convert from inches to feet:

$20 \div 12 = \frac{5}{3}$ feet

$15 \div 12 = \frac{5}{4}$ feet

Now find half their product:

$A = \frac{1}{2} \times \frac{5}{3}\times \frac{5}{4} = \frac{25}{24} = 1 \frac{1}{24}$ square feet

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Example Question #3 : How To Find The Area Of A Triangle

Rectangles

Note: Figure NOT drawn to scale.

What percent of the above figure is green?

Possible Answers:

$62\frac{1}{2} \%$

The correct answer is not given among the other choices.

$78 \frac{3}{4} \%$

$68 \frac{8}{9} \%$

$73 \frac{1}{3} \%$

Correct answer:

$73 \frac{1}{3} \%$

Explanation:

The area of the entire rectangle is the product of its length and width, or

$120 \times 50 = 6,000$ .

The area of the right triangle is half the product of its legs, or

$\frac{1}{2} \times 40 \times 80 = 1,600$

The area of the green region is therefore the difference of the two, or

6,000 - 1,600 = 4,400 .

The green region is therefore

$\frac{4,400}{6,000} \times 100 = 73 \frac{1}{3} \%$

of the rectangle.

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Example Question #1 : How To Find The Area Of A Triangle

Rectangles

Note: Figure NOT drawn to scale.

Refer to the above diagram. Give the ratio of the area of the green region to that of the white region.

Possible Answers:

2:1

11:4

21:9

The correct answer is not given among the other choices.

23:7

Correct answer:

11:4

Explanation:

The area of the entire rectangle is the product of its length and width, or

$120 \times 50 = 6,000$ .

The area of the right triangle is half the product of its legs, or

$\frac{1}{2} \times 40 \times 80 = 1,600$

The area of the green region is therefore the difference of the two, or

6,000 - 1,600 = 4,400 .

The ratio of the area of the green region to that of the white region is

$\frac{4,400}{1,600} = \frac{4,400 \div 400}{1,600\div 400} = \frac{11}{4}$

That is, 11 to 4.

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Example Question #1 : Area Of A Triangle

A triangle has a height of 9 inches and a base that is one third as long as the height. What is the area of the triangle, in square inches?

Possible Answers:

$\frac{27}{2}$

None of these

Correct answer:

$\frac{27}{2}$

Explanation:

The area of a triangle is found by multiplying the base times the height, divided by 2.

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

Given that the height is 9 inches, and the base is one third of the height, the base will be 3 inches.

$b=h\div3=9\div3=3$

We now have both the base (3) and height (9) of the triangle. We can use the equation to solve for the area.

$\text{Area} =\frac{9\times 3}{2}$

$\text{Area} =\frac{27}{2}$

The fraction cannot be simplified.

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Example Question #51 : Geometry

The hypotenuse of a right triangle is feet; it has one leg feet long. Give its area in square inches.

Possible Answers:

$42,120 \textrm{ in}^{2}$

$43,120 \textrm{ in}^{2}$

$101,088 \textrm{ in}^{2}$

$38,880 \textrm{ in}^{2}$

$77,760 \textrm{ in}^{2}$

Correct answer:

$38,880 \textrm{ in}^{2}$

Explanation:

The area of a right triangle is half the product of the lengths of its legs, so we need to use the Pythagorean Theorem to find the length of the other leg. Set c = 39 , b = 36 :

$a^{2} = c^{2} - b^{2}$

$a^{2} = 39 ^{2} - 36^{2}$

$a^{2} = 1,521 - 1,296$

$a^{2} = 225$

$a = \sqrt{225} = 15$

The legs have length and feet; multiply both dimensions by to convert to inches:

$15 \times 12 = 180$ inches

$36 \times 12 = 432$ inches.

Now find half the product:

$A = \frac{1}{2} \cdot 180 \cdot 432 = 38,880\ in^{2}$

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Example Question #26 : Area Of A Triangle

What is the area (in square feet) of a triangle with a base of feet and a height of feet?

Possible Answers:

Correct answer:

Explanation:

The area of a triangle is found by multiplying the base times the height, divided by .

$Area =\frac{base\cdot height}{2}$

$Area =\frac{4\cdot 7}{2}$

Area=14

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Example Question #101 : Geometry

What is the area of a triangle with a base of and a height of ?

Possible Answers:

Correct answer:

Explanation:

The formula for the area of a triangle is $\dpi{100} Area=\frac{1}{2}\times base\times height$ .

Plug the given values into the formula to solve:

$\dpi{100} Area=\frac{1}{2}\times 12\times 3$

$\dpi{100} Area=\frac{1}{2}\times 36$

$\dpi{100} Area=18$

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Example Question #11 : How To Find The Area Of A Triangle

Right triangle 2

Give the perimeter of the above triangle in feet.

Possible Answers:

$3 \frac{1}{3}\textrm{ ft }$

$2 \frac{2}{3}\textrm{ ft }$

$3 \frac{1}{2}\textrm{ ft }$

$2 \frac{1}{2}\textrm{ ft }$

Correct answer:

$2 \frac{1}{2}\textrm{ ft }$

Explanation:

The perimeter of the triangle - the sum of the lengths of its sides - is

5 + 12 + 13 = 30 inches.

Divide by 12 to convert to feet:

$30 \div 12 = 2 \textrm{ R }6$

As a fraction, this is $2 \frac{6}{12}$ or $2 \frac{1}{2}$ feet,

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Example Question #61 : Geometry

Rectangles 3

The above diagram shows Rectangle RECT , with midpoint of $\overline{CT}$ .

The area of $\bigtriangleup ECM$ is 225. Evaluate

Possible Answers:

Correct answer:

Explanation:

is the midpoint of $\overline{CT}$ , so $\bigtriangleup ECM$ has as its base $CM= \frac{1}{2} \cdot CT = \frac{1}{2} \cdot 2x =x$ ; its height is $EC = \frac{1}{2} x$ .

Its area is half their product, or

$\frac{1}{2} \cdot x \cdot \frac{1}{2} x = \frac{1}{4} x^{2}$

Set this equal to 225:

$\frac{1}{4} x^{2} = 225$

$4 \cdot \frac{1}{4} x^{2} = 4 \cdot 225$

$x^{2} = 900$

$x = \sqrt{900}= 30$ .

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Example Question #62 : Geometry

Find the area of a triangle with a height of 12in and a base that is half the height.

Possible Answers:

$18\text{in}^2$

$24\text{in}^2$

$6\text{in}^2$

$36\text{in}^2$

$30\text{in}^2$

Correct answer:

$36\text{in}^2$

Explanation:

To find the area of a triangle, we will use the following formula:

$A = \frac{1}{2} \cdot b \cdot h$

where b is the base and h is the height of the triangle.

Now, we know the height of the triangle is 12in. We also know the base of the triangle is half of the height. Therefore, the base of the triangle is 6in.

So, we can substitute. We get

$A = \frac{1}{2} \cdot 6\text{in} \cdot 12\text{in}$

$A = 3\text{in} \cdot 12\text{in}$

$A = 36\text{in}^2$

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SSAT Middle Level Math : Plane Geometry

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

All SSAT Middle Level Math Resources

Example Questions

Example Question #41 : Geometry

Example Question #3 : How To Find The Area Of A Triangle

Example Question #1 : How To Find The Area Of A Triangle

Example Question #1 : Area Of A Triangle

Example Question #51 : Geometry

Example Question #26 : Area Of A Triangle

Example Question #101 : Geometry

Example Question #11 : How To Find The Area Of A Triangle

Example Question #61 : Geometry

Example Question #62 : Geometry

All SSAT Middle Level Math Resources

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