How to find the area of a triangle

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ISEE Middle Level Quantitative Reasoning › How to find the area of a triangle

Questions 1 - 10

What is the area of the triangle?

Question_11

$\small 35$

$\small 70$

$\small 84$

$\small 42$

Explanation

Area of a triangle can be determined using the equation:

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

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

In an equilateral triangle, which of the following is NOT true?

There is a 90 degree angle

All sides are equal

All angles are equal

There is a 60 degree angle

Explanation

In an equilateral triangle, all sides and angles are equal. All the angles equal 60 degrees, so there is a 60 degree angle.

Therefore, the answer choice, “There is a 90 degree angle” is not true and is the correct answer choice.

Find the area of a triangle with a base of 8in and a height of 12in.

$48\text{in}^2$

$96\text{in}^2$

$24\text{in}^2$

$56\text{in}^2$

$36\text{in}^2$

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 base of the triangle is 8in. We also know the height is 12in. So, we can substitute. We get

$A = \frac{1}{2} \cdot 8\text{in} \cdot 12\text{in}$

$A = 4\text{in} \cdot 12\text{in}$

$A = 48\text{in}^2$

Find the area of a triangle with a base of 17in and a height of 12in.

$102\text{in}^2$

$41\text{in}^2$

$29\text{in}^2$

$144\text{in}^2$

$121\text{in}^2$

Explanation

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

$\text{area of triangle} = \frac{1}{2} \cdot b \cdot h$

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

Now, we know the base of the triangle is 17in. We also know the height of the triangle is 12in.

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

$\text{area of triangle} = \frac{1}{2} \cdot 17\text{in} \cdot 12\text{in}$

$\text{area of triangle} = 17\text{in} \cdot 6\text{in}$

$\text{area of triangle} = 102\text{in}^2$

A triangle has base 80 inches and area 4,200 square inches. What is its height?

$105 ; \textrm{in}$

$210 ; \textrm{in}$

$140 ; \textrm{in}$

$160 ; \textrm{in}$

$120 ; \textrm{in}$

Explanation

Use the area formula for a triangle, setting :

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

$4200 = \frac{1}{2}\cdot 80 \cdot h = 40h$

$h = 4200 \div 40 = 105$ inches

Pentagon

Figure NOT drawn to scale

Square has area 1,600. ; . Which of the following is the greater quantity?

(a) The area of $\bigtriangleup AZX$

(b) The area of $\bigtriangleup BZY$

(b) is the greater quantity

It is impossible to determine which is greater from the information given

(a) is the greater quantity

(a) and (b) are equal

Explanation

Square has area 1,600, so the length of each side is $\sqrt{1,600 }= 40$ .

Since ,

Therefore, .

$\bigtriangleup AZX$ has as its area $\frac{1}{2} \cdot AX \cdot AZ$ ; $\bigtriangleup BZY$ has as its area $\frac{1}{2} \cdot BX \cdot BZ$ .

Since and , it follows that

$BX \cdot BZ > AX \cdot AZ$

and

$\frac{1}{2} \cdot BX \cdot BZ >\frac{1}{2} \cdot AX \cdot AZ$

$\bigtriangleup BZY$ has greater area than $\bigtriangleup AZX$ .

Square 1

Figure NOT drawn to scale.

In the above diagram, Square has area 400. Which is the greater quantity?

(a) The area of $\bigtriangleup SQE$

(b) The area of $\bigtriangleup UAR$

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

It is impossible to determine which is greater from the information given

Explanation

Square has area 400, so its common sidelength is the square root of 400, or 20. Therefore,

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

$\bigtriangleup SQE$ has legs $\overline{SQ}$ and $\overline{SE}$ , so its area is

$\frac{1}{2} \cdot SQ \cdot SE = \frac{1}{2} \cdot 9 \cdot 20 = 90$ .

$\bigtriangleup UAR$ has legs $\overline{UA}$ and $\overline{AR}$ , so its area is

$\frac{1}{2} \cdot AR \cdot UA = \frac{1}{2} \cdot 11 \cdot 20 = 110$ .

$\bigtriangleup UAR$ has the greater area.

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.

The correct answer is not given among the other choices.

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

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.

Find the area of a triangle with the following measurements:

base = 8in
height = 7in

$28\text{in}^2$

$56\text{in}^2$

$36\text{in}^2$

$24\text{in}^2$

$15\text{in}^2$

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 base is 8in and the height is 7in. Knowing this, we can substitute into the formula. So, we get

$A = \frac{1}{2} \cdot 8\text{in} \cdot 7\text{in}$

$A = 4\text{in} \cdot 7\text{in}$

$A = 28\text{in}^2$

The sum of the lengths of the legs of an isosceles right triangle is one meter. What is its area in square centimeters?

$1,250 \textrm{ cm} ^{2}$

It is impossible to determine the area from the information given

$2,500 \textrm{ cm} ^{2}$

$250 \textrm{ cm} ^{2}$

$125 \textrm{ cm} ^{2}$

Explanation

The legs of an isosceles right triangle have equal length, so, if the sum of their lengths is one meter, which is equal to 100 centimeters, each leg measures half of this, or

$100 \div 2 = 50$ centimeters.

The area of a triangle is half the product of its height and base; for a right triangle, the legs serve as height and base, so the area of the triangle is

$\frac{1}{2} \times 50 \times 50 = 1,250$ square centimeters.

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