How to find the area of an acute / obtuse triangle - SSAT Upper Level Quantitative (Math)

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Question

Give the area of the above triangle.

Answer

By Heron's formula, the area of a triangle given its sidelengths is

$A =\sqrt{ s (s-a) (s-b) (s-c)}$ ,

where are the sidelengths and

$s = \frac{1}{2} (a+b+c)$ ,

or half the perimeter.

Setting ,

$s = \frac{1}{2} (a+b+c) = \frac{1}{2} (16+20+24) = 30$ .

Therefore,

$A =\sqrt{ s (s-a) (s-b) (s-c)}$

$A =\sqrt{ 30 (30-16) (30-20) (30-24)}$

$A =\sqrt{ 30 \cdot 14 \cdot 10 \cdot 6 }$

$A = \sqrt{25,200 }$

$A = \sqrt{3,600 } \cdot\sqrt{7}$

$A = 60 \sqrt{7 }$

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Question

Give the area of the above triangle.

Answer

By Heron's formula, the area of a triangle given its sidelengths is

$A =\sqrt{ s (s-a) (s-b) (s-c)}$

Where are the sidelengths and

$s = \frac{1}{2} (a+b+c)$ ,

or half the perimeter.

Setting ,

$s = \frac{1}{2} (a+b+c) = \frac{1}{2} (50+45+85) = 90$

Therefore,

$A =\sqrt{ s (s-a) (s-b) (s-c)}$

$A =\sqrt{90 (90 -50) (90 -45) (90 -85)}$

$A =\sqrt{90 \cdot 40 \cdot 45 \cdot 5 }$

$A =\sqrt{90 \cdot 40 \cdot 45 \cdot 5 }$

$A =\sqrt{810,000 }$

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Question

The base of a triangle is inches, and the height of the triangle is inches. In terms of , what is the area of the triangle?

Answer

Find the area of the triangle by using the formula $\frac{base\times height}{2}$ .

Now, substitute in for the base and for the height.

$\text{Area}=\frac{y\times8}{2}=\frac{8y}{2}=4y$

Don't forget to include the units,

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Question

The base of the triangle is . The height of the triangle is a multiple of between and . What is the area of the triangle?

Answer

First, find the height of the triangle by listing out the multiples of .

Since is the only multiple of that is between and , it must be the height.

Now, find the area of the triangle.

$\text{Area}=\frac{5\times24}{2}=60$

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Question

The base of an obtuse triangle is , and the height is . What is the area of the triangle?

Answer

Use the following formula to find the area of a triangle:

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

Now, substitute in for the base and for the height.

$\text{Area}=\frac{20\times 12}{2}=\frac{240}{2}=120$

The area of the triangle is .

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Question

The height of an acute triangle is , and the base is . What is the area of this triangle?

Answer

Use the following formula to find the area of a triangle:

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

Now, substitute in for the base and for the height.

$\text{Area}=\frac{40\times14}{2}=\frac{560}{2}=280$

The area of the triangle is .

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Question

The base of a triangle is , and the height is . In terms of , what is the area of the triangle?

Answer

Use the following formula to find the area of a triangle:

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

Now, substitute in for the base and for the height.

$\text{Area}=\frac{4t\times 5t}{2}=\frac{20t^2}{2}=10t^2$

The area of the triangle is .

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Question

The base of a triangle is , and the height of the triangle is . If the area of the triangle is , what is the value of ?

Answer

Use the following formula to find the area of a triangle:

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

Now, substitute in for the base, for the height, and for the area..

Use algebraic opertions to solve for x.

$120=\frac{12x\times5}{2}$

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Question

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

Answer

Use the following formula to find the area of a triangle:

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

Now, substitute in for the base and for the height.

$\text{Area}=\frac{4\times20}{2}=\frac{80}{2}=40$

The area of the triangle is .

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Question

The height of a triangle is , and the base is . In terms of , what is the area of the triangle?

Answer

Use the following formula to find the area of a triangle:

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

Now, substitute in for the base and for the height.

$\text{Area}=\frac{6a\times8a}{2}=\frac{48a^2}{2}=24a^2$

The area of the triangle is .

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Question

The height of the triangle is , and the base of the triangle is . If the area of the triangle is , what is the value of ?

Answer

Use the formula for the area of a triangle.

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

Substitute in for height, for the base, and for the area.

From here, use algebraic operations to isolate d on one side and all other numbers on the other side.

$80=\frac{5d\times 8}{2}$

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Question

The height of a triangle is , and the base of the triangle is . In terms of , what is the area of the triangle?

Answer

Use the following formula to find the area of a triangle:

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

Now, substitute in for the base and for the height.

$\text{Area}=\frac{9y\times6y}{2}=\frac{54y^2}{2}=27y^2$

The area of the triangle is .

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Question

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

Answer

Use the following formula to find the area of a triangle:

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

Now, substitute in for the base and for the height.

$\text{Area}=\frac{4\times14}{2}=\frac{56}{2}=28$

The area of the triangle is .

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Question

The height of a triangle is , and the base is . In terms of , what is the area of the triangle?

Answer

Use the following formula to find the area of a triangle:

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

Now, substitute in for the base and for the height.

$\text{Area}=\frac{6g\times10g}{2}=\frac{60g^2}{2}=30g^2$

The area of the triangle is .

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