Properties of Triangles - SSAT Upper Level Quantitative

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Question

Give the area of the above triangle.

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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.

Tap to reveal answer

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?

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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?

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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?

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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?

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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?

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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 ?

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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?

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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?

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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.

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 ?

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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?

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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.

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?

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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?

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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.

The area of the triangle is .

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Question

The perimeter of a triangle is . If one side has the length , and another side has the length , what is the length of the third side?

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Answer

The perimeter is the length of all the sides added up.

$\text{Perimeter}$=$\text{side 1 + side 2 + side 3}$

Using the information given in the question,

$18a=8a+6a+$\text{side 3}$$

Now, solve for side 3.

$$\text{Side 3}$=18a-8a-6a=4a$

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Question

If the perimeter of the triangle is and two of the sides are given in the figure below, what is the length of the third side?

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Answer

To find the perimeter of a triangle, add up all of its sides.

Let be the length of the third side.

The length of the third side is .

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Question

The perimeter of the triangle is , and two of the sides are already given. What is the length of the third side?

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Answer

Add up all the sides to find the perimeter of the triangle.

Let be the length of the third side.

The length of the third side is .

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Question

The perimeter of the triangle is , and two of the sides are already given. What is the length of the third side?

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Answer

Add up all the sides to find the perimeter of the triangle.

Let be the length of the third side.

The length of the third side is .

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Question

The perimeter of the triangle is , and two of the sides are given. What is the length of the third side?

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Answer

Add up all the sides to find the perimeter of the triangle.

Let be the length of the third side.

The length of the third side is .

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Question

The perimeter of the triangle is , and two of the sides are given. What is the length of the third side?

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Answer

Add up all the sides to find the perimeter of the triangle.

Let be the length of the third side.

The length of the third side is .

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