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

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

The area of an isosceles right triangle is $80\:in^2$ . What is its height that is correlative and perpendicular to a side that is not the hypotenuse?

Possible Answers:

$\frac{8}{5}\sqrt{26}\:in$

$3\sqrt{5}\:in$

$4\sqrt{10}\:in$

$10\sqrt{2}\:in$

$4\sqrt{5}\:in$

Correct answer:

$4\sqrt{10}\:in$

Explanation:

Recall that an isosceles right triangle is a 45-45-90 triangle. That means that it looks like this:

_tri11

This makes calculating the area very easy! Recall, the area of a triangle is defined as:

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

However, since b=h=x for our triangle, we know:

$A=\frac{1}{2}x^2$

Now, we know that $A=80\:in^2$ . Therefore, we can write:

$80=\frac{1}{2}x^2$

Solving for , we get:

160=x^2

$x=\sqrt{160}= \sqrt{16}*\sqrt{10}=4\sqrt{10}\:in$

This is the length of the height of the triangle for the side that is not the hypotenuse.

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

What is the area of an isosceles right triangle that has an hypotenuse of length $18\:cm$ ?

Possible Answers:

$81\:cm^2$

$324\:cm^2$

$40.5\:cm^2$

$42\:cm^2$

$121.5\:cm^2$

Correct answer:

$81\:cm^2$

Explanation:

Based on the information given, you know that your triangle looks as follows:

_tri21

This is a 45-45-90 triangle. Recall your standard triangle:

Triangle454590

You can set up the following ratio between these two figures:

$\frac{x}{1}=\frac{18}{\sqrt{2}}$

Now, the area of the triangle will merely be $\frac{1}{2}x^2$ (since both the base and the height are ). For your data, this is:

$A=\frac{1}{2}* \frac{18}{\sqrt{2}} * \frac{18}{\sqrt{2}}=\frac{18*18}{4}=9*9=81\:cm^2$

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Example Question #191 : Plane Geometry

Find the height of an isoceles right triangle whose hypotenuse is $5\sqrt2$

Possible Answers:

$10\sqrt2$

$25\sqrt2$

$5\sqrt2$

Correct answer:

Explanation:

To solve simply realize the hypotenuse of one of these triangles is of the form $s\sqrt2$ where s is side length. Thus, our answer is .

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Example Question #1 : How To Find The Height Of A 45/45/90 Right Isosceles Triangle

The area of an isosceles right triangle is $100\:cm^2$ . What is its height that is correlative and perpendicular to this triangle's hypotenuse?

Possible Answers:

$10\sqrt{5}\:cm$

$10\sqrt{2}\:cm$

$5\sqrt{2}\:cm$

$20\:cm$

$10\:cm$

Correct answer:

$10\:cm$

Explanation:

Recall that an isosceles right triangle is a 45-45-90 triangle. That means that it looks like this:

_tri11

This makes calculating the area very easy! Recall, the area of a triangle is defined as:

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

However, since b=h=x for our triangle, we know:

$A=\frac{1}{2}x^2$

Now, we know that $A=100\:cm^2$ . Therefore, we can write:

$100=\frac{1}{2}x^2$

Solving for , we get:

200=x^2

$x=\sqrt{200}= \sqrt{2}*\sqrt{100}=10\sqrt{2}$

However, be careful! Notice what the question asks: "What is its height that is correlative and perpendicular to this triangle's hypotenuse?" First, let's find the hypotenuse of the triangle. Recall your standard 45-45-90 triangle:

_tri51

Since one of your sides is $10\sqrt{2}$ , your hypotenuse is $10\sqrt{2} * \sqrt{2}=20\:cm$ .

Okay, what you are actually looking for is in the following figure:

_tri61

Therefore, since you know the area, you can say:

$100=\frac{1}{2}\ast h*20$

Solving, you get: h=10 .

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ACT Math : How to find the height 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 #191 : Plane Geometry

Example Question #1 : How To Find The Height Of A 45/45/90 Right Isosceles Triangle

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