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ACT Math : How to find the area of an acute / obtuse isosceles triangle

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Example Question #1 : Acute / Obtuse Isosceles Triangles

What is the area of an isosceles triangle with a vertex of 120 $\displaystyle 120$ degrees and two sides equal to $4\:cm$ $\displaystyle 4\:cm$ ?

Possible Answers:

$12\sqrt{3}\:cm^2$ $\displaystyle 12\sqrt{3}\:cm^2$

$15\sqrt{2}\:cm^2$ $\displaystyle 15\sqrt{2}\:cm^2$

$20\:cm^2$ $\displaystyle 20\:cm^2$

$20\sqrt{3}\:cm^2$ $\displaystyle 20\sqrt{3}\:cm^2$

$4\sqrt{3}\:cm^2$ $\displaystyle 4\sqrt{3}\:cm^2$

Correct answer:

$4\sqrt{3}\:cm^2$ $\displaystyle 4\sqrt{3}\:cm^2$

Explanation:

Based on the description of your triangle, you can draw the following figure:

_tri71

You can do this because you know:

The two equivalent sides are given.
Since a triangle is $\displaystyle 180$ degrees, you have only $\displaystyle 180-120$ or $\displaystyle 60$ degrees left for the two angles of equal size. Therefore, those two angles must be $\displaystyle 30$ degrees and $\displaystyle 30$ degrees.

Now, based on the properties of an isosceles triangle, you can draw the following as well:

_tri72

Based on your standard reference $\displaystyle 30-60-90$ triangle, you know:

$\frac{h}{4}=\frac{1}{2}$ $\displaystyle \frac{h}{4}=\frac{1}{2}$

Therefore, $\displaystyle h$ is $\displaystyle 2$ .

This means that $\displaystyle x$ is $2\sqrt{3}$ $\displaystyle 2\sqrt{3}$ and the total base of the triangle is $4\sqrt{3}$ $\displaystyle 4\sqrt{3}$ .

Now, the area of the triangle is:

$\frac{1}{2}bh$ $\displaystyle \frac{1}{2}bh$ or $\frac{1}{2}*2*4\sqrt{3}=4\sqrt{3}\:cm^2$ $\displaystyle \frac{1}{2}*2*4\sqrt{3}=4\sqrt{3}\:cm^2$

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Example Question #1 : Acute / Obtuse Isosceles Triangles

An isosceles triangle has a height of $\displaystyle 4$ and a base of $\displaystyle 7$ . What is its area?

Possible Answers:

$\displaystyle 24$

$\displaystyle 28$

$\displaystyle 17$

$\displaystyle 14$

Correct answer:

$\displaystyle 14$

Explanation:

Use the formula for area of a triangle:

$A=\frac{1}{2}(base)(height)$ $\displaystyle A=\frac{1}{2}(base)(height)$

$A=\frac{1}{2}(4)(7)=14$ $\displaystyle A=\frac{1}{2}(4)(7)=14$

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Example Question #2 : Acute / Obtuse Isosceles Triangles

An isosceles triangle has a base length of $\displaystyle 10$ and a height that is twice its base length. What is the area of this triangle?

Possible Answers:

150 $\displaystyle 150$

200 $\displaystyle 200$

$\displaystyle 50$

100 $\displaystyle 100$

Correct answer:

100 $\displaystyle 100$

Explanation:

1. Find the height of the triangle:

$height=2\cdot base$ $\displaystyle height=2\cdot base$

$\displaystyle height=2(10)=20$

2. Use the formula for area of a triangle:

$A=\frac{1}{2}(b)(h)$ $\displaystyle A=\frac{1}{2}(b)(h)$

$A=\frac{1}{2}(10)(20)=100$ $\displaystyle A=\frac{1}{2}(10)(20)=100$

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ACT Math : How to find the area of an acute / obtuse isosceles triangle

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #1 : Acute / Obtuse Isosceles Triangles

Example Question #1 : Acute / Obtuse Isosceles Triangles

Example Question #2 : Acute / Obtuse Isosceles Triangles

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