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ACT Math : How to find the area of a 45/45/90 right isosceles triangle

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Example Question #1 : 45/45/90 Right Isosceles Triangles

What is the area of an isosceles right triangle with a hypotenuse of $26\:cm$ ?

Possible Answers:

$169\sqrt{2}\:cm^2$

$26\sqrt{2}\:cm^2$

$169\:cm^2$

$676\:cm^2$

$13\sqrt{2}\:cm^2$

Correct answer:

$169\:cm^2$

Explanation:

Now, this is really your standard 45-45-90 triangle. Since it is a right triangle, you know that you have at least one -degree angle. The other two angles must each be degrees, because the triangle is isosceles.

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

_tri101

This is derived from your reference triangle for the 45-45-90 triangle:

Triangle454590

For our triangle, we could call one of the legs . We know, then:

$\frac{x}{26}=\frac{1}{\sqrt{2}}$

Thus, $x=\frac{26}{\sqrt{2}}$ .

The area of your triangle is:

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

For your data, this is:

$\frac{1}{2}*\frac{26}{\sqrt{2}}*\frac{26}{\sqrt{2}}=\frac{26^2}{4}=169\:cm^2$

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Example Question #2 : 45/45/90 Right Isosceles Triangles

What is the area of an isosceles right triangle with a hypotenuse of $9\sqrt{2}\:cm$ ?

Possible Answers:

$54\sqrt{2}\:cm^2$

$162\:cm^2$

$81\:cm^2$

$40.5\:cm^2$

$18\sqrt{2}\:cm^2$

Correct answer:

$40.5\:cm^2$

Explanation:

Now, this is really your standard 45-45-90 triangle. Since it is a right triangle, you know that you have at least one -degree angle. The other two angles must each be degrees because the triangle is isosceles.

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

_tri111

This is derived from your reference triangle for the 45-45-90 triangle:

Triangle454590

For our triangle, we could call one of the legs . We know, then:

$\frac{x}{1}=\frac{9\sqrt{2}}{\sqrt{2}}$

Thus, x=9 .

The area of your triangle is:

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

For your data, this is:

$\frac{1}{2}*9*9=40.5\:cm^2$

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Example Question #3 : 45/45/90 Right Isosceles Triangles

What is the area of an isosceles right triangle with a hypotenuse of $7\sqrt{2}\:cm$ ?

Possible Answers:

$98\:cm^2$

$49\sqrt{2}\:cm^2$

$73.5\:cm^2$

$49\:cm^2$

$24.5\:cm^2$

Correct answer:

$24.5\:cm^2$

Explanation:

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

_tri121

This is derived from your reference triangle for the 45-45-90 triangle:

Triangle454590

For our triangle, we could call one of the legs . We know, then:

$\frac{x}{1}=\frac{7\sqrt{2}}{\sqrt{2}}$

Thus, x=7 .

The area of your triangle is:

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

For your data, this is:

$\frac{1}{2}*7*7=24.5\:cm^2$

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Example Question #4 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

$\Delta PQR$ is a right isosceles triangle with hypotenuse . What is the area of $\Delta PQR$ ?

Possible Answers:

$330\sqrt2\textup{ units}^{2}$

$640\textup{ units}^{2}$

$529\textup{ units}^{2}$

$723\textup{ units}^{2}$

$299\sqrt2\textup{ units}^{2}$

Correct answer:

$529\textup{ units}^{2}$

Explanation:

Right isosceles triangles (also called "45-45-90 right triangles") are special shapes. In a plane, they are exactly half of a square, and their sides can therefore be expressed as a ratio equal to the sides of a square and the square's diagonal:

$x: x: x\sqrt{2}$ , where $x\sqrt{2}$ is the hypotenuse.

In this case, maps to $x\sqrt{2}$ , so to find the length of a side (so we can use the triangle area formula), just divide the hypotenuse by $\sqrt2$ :

$\frac{46}{\sqrt2} = \frac{46}{\sqrt2} \cdot (\frac{\sqrt2}{\sqrt2}) = \frac{46\sqrt{2}}{2} = 23\sqrt{2}$

So, each side of the triangle is $23\sqrt2$ long. Now, just follow your formula for area of a triangle:

$A\Delta = \frac{lh}{2} = \frac{(23\sqrt{2})(23\sqrt{2})}{2} = \frac{1058}{2} = 529$

Thus, the triangle has an area of $529\textup{ units}^{2}$ .

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ACT Math : How to find the area of a 45/45/90 right isosceles triangle

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : 45/45/90 Right Isosceles Triangles

Example Question #2 : 45/45/90 Right Isosceles Triangles

Example Question #3 : 45/45/90 Right Isosceles Triangles

Example Question #4 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

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