Acute / Obtuse Triangles - SSAT Upper Level Quantitative

Card 0 of 340

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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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?

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?

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?

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?

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?

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?

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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