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ISEE Middle Level Math : Plane Geometry

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Example Question #31 : Geometry

Find the perimeter of an equilateral triangle with a side of length 7in.

Possible Answers:

$28\text{in}$

$21\text{in}$

$24\text{in}$

$29\text{in}$

$14\text{in}$

Correct answer:

$21\text{in}$

Explanation:

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

P = a+b+c

where a, b, and c are the lengths of the sides of the triangle.

Now, we know the equilateral triangle has one side of length 7in. Because it is an equilateral triangle, all sides are equal. Therefore, all sides are 7in. So, we can substitute. We get

$P = 7\text{in} + 7\text{in} + 7\text{in}$

$P = 21\text{in}$

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Example Question #31 : How To Find The Perimeter Of A Triangle

Find the perimeter of an equilateral triangle with a base of 13in.

Possible Answers:

$39\text{in}$

$27\text{in}$

$48\text{in}$

$36\text{in}$

$169\text{in}$

Correct answer:

$39\text{in}$

Explanation:

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

P = a+b+c

where a, b, and c are the lengths of the sides of the triangle.

Now, we know the base of the triangle is 13in. Because it is an equilateral triangle, all sides are equal. Therefore, all sides are 13in. So, we get

$P = 13\text{in} + 13\text{in} + 13\text{in}$

$P = 39\text{in}$

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Example Question #33 : Geometry

Find the perimeter of an equilateral triangle with a base of length 9in.

Possible Answers:

$\text{There is not enough information to solve the problem.}$

$27\text{in}$

$36\text{in}$

$32\text{in}$

$18\text{in}$

Correct answer:

$27\text{in}$

Explanation:

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

P = a+b+c

where a, b, and c are the lengths of the sides of the triangle.

Now, we know the base of the triangle has a length of 9in. Because it is an equilateral triangle, all lengths are the same. Therefore, all lengths are 9in.

Knowing this, we can substitute into the formula. We get

$P = 9\text{in} +9\text{in} +9\text{in}$

$P = 27\text{in}$

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

A triangle has a base of $\small 43$ and an area of $\small 2042.5$ . What is the height?

Possible Answers:

$\small 190$

$\small 93$

$\small 48$

$\small 95$

$\small 47$

Correct answer:

$\small 95$

Explanation:

The area of a triangle is found by multiplying the base by the height and dividing by two:

$\small \frac{b\cdot h}{2}$

In this problem we are given the base, which is $\small 43$ , and the area, which is $\small 2042.5$ . First we write an equation using $\small h$ as our variable.

$\small \frac{43\cdot h}{2}=2042.5$

To solve this equation, first multply both sides by $\small 2$ , becuase multiplication is the opposite of division and therefore allows us to eliminate the $\small 2$ .

The left-hand side simplifies to:

$\small \frac{43\cdot h}{2}\cdot 2=43\cdot h$

The right-hand side simplifies to:

$\small 2042.5\cdot 2=4085$

So our equation is now:

$\small \small 43\cdot h=4085$

Next we divide both sides by $\small \small 43$ , because division is the opposite of multiplication, so it allows us to isolate the variable by eliminating $\small \small 43$ .

$\small \frac{43\cdot h}{43}= h$

$\small \frac{4085}{43}=95$

$\small h=95$

So the height of the triangle is $\small 95$ .

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

Triangle

Note: Figure NOT drawn to scale.

The above triangle has area 36 square inches. If $x = 4.5 \textrm{ in}$ , then what is ?

Possible Answers:

$y = 24\textrm{ in}$

$y = 16 \textrm{ in}$

$y = 26 \textrm{ in}$

$y = 32 \textrm{ in}$

$y = 20 \textrm{ in}$

Correct answer:

$y = 16 \textrm{ in}$

Explanation:

The area of a triangle is one half the product of its base and its height - in the above diagram, that means

$A = \frac{1}{2}xy$ .

Substitute A = 36, x = 4.5 , and solve for .

$\frac{1}{2} \cdot 4.5 \cdot y = 36$

2.25 y = 36

$2.25 y \div 2.25= 36\div 2.25$

$y = 16 \textrm{ in}$

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Example Question #4 : Area Of A Triangle

Please use the following shape for the question. 5x3-adams-graphoc

What is the area of this shape?

Possible Answers:

$25\ in^{2}$

$40\ in^{2}$

$21\ in^{2}$

$15\ in^{2}$

$32.5\ in^{2}$

Correct answer:

$32.5\ in^{2}$

Explanation:

From this shape we are able to see that we have a square and a triangle, so lets split it into the two shapes to solve the problem. We know we have a square based on the 90 degree angles placed in the four corners of our quadrilateral.

Since we know the first part of our shape is a square, to find the area of the square we just need to take the length and multiply it by the width. Squares have equilateral sides so we just take 5 times 5, which gives us 25 inches squared.

We now know the area of the square portion of our shape. Next we need to find the area of our right triangle. Since we know that the shape below the triangle is square, we are able to know the base of the triangle as being 5 inches, because that base is a part of the square's side.

To find the area of the triangle we must take the base, which in this case is 5 inches, and multipy it by the height, then divide by 2. The height is 3 inches, so 5 times 3 is 15. Then, 15 divided by 2 is 7.5.

We now know both the area of the square and the triangle portions of our shape. The square is 25 inches squared and the triangle is 7.5 inches squared. All that is remaining is to added the areas to find the total area. Doing this gives us 32.5 inches squared.

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Example Question #5 : Area Of A Triangle

What is the area of the triangle?

Question_11

Possible Answers:

$\small 35$

$\small 84$

$\small 42$

$\small 70$

Correct answer:

$\small 35$

Explanation:

Area of a triangle can be determined using the equation:

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

$\small A=\frac{1}{2}(14)(5)=35$

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Example Question #1 : Area Of A Triangle

Bill paints a triangle on his wall that has a base parallel to the ground that runs from one end of the wall to the other. If the base of the wall is 8 feet, and the triangle covers 40 square feet of wall, what is the height of the triangle?

Possible Answers:

$11\: feet$

$20\: feet$

$8\: feet$

$9\: feet$

$10\: feet$

Correct answer:

$10\: feet$

Explanation:

In order to find the area of a triangle, we multiply the base by the height, and then divide by 2.

$\small \frac{b\cdot h}{2}$

In this problem we are given the base and the area, which allows us to write an equation using $\small h$ as our variable.

$\small \frac{8\cdot h}{2}=40$

Multiply both sides by two, which allows us to eliminate the two from the left side of our fraction.

The left-hand side simplifies to:

$\small \frac{8\cdot h}{2}\cdot 2=8\cdot h$

The right-hand side simplifies to:

$\small 40\cdot 2=80$

Now our equation can be rewritten as:

$\small 8\cdot h=80$

Next we divide by 8 on both sides to isolate the variable:

$\small \frac{8\cdot h}{8}=h$

$\small \frac{80}{8}=10$

$\small h=10$

Therefore, the height of the triangle is $\small 10 \: feet$ .

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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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ISEE Middle Level Math : Plane Geometry

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

All ISEE Middle Level Math Resources

Example Questions

Example Question #31 : Geometry

Example Question #31 : How To Find The Perimeter Of A Triangle

Example Question #33 : Geometry

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

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

Example Question #4 : Area Of A Triangle

Example Question #5 : Area Of A Triangle

Example Question #1 : Area Of A Triangle

Example Question #41 : Geometry

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

All ISEE Middle Level Math Resources

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