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Basic Geometry : 45/45/90 Right Isosceles Triangles

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Example Question #11 : How To Find The Height Of A 45/45/90 Right Isosceles Triangle

If the hypotenuse of an isoceles right triangle is 144 , what is the length of the height?

Possible Answers:

$72\sqrt2$

$24\sqrt3$

$88\sqrt2$

Correct answer:

$72\sqrt2$

Explanation:

An isoceles right triangle is another way of saying that the triangle is a 45-45-90 triangle.

Now, recall the Pythagorean theorem:

$\text{base}^2+\text{height}^2=\text{hypotenuse}^2$

Because we are working with a 45-45-90 triangle, the base and the height have the same length. We can rewrite the above equation as the following:

$\text{height}^2+\text{height}^2=\text{hypotenuse}^2$

$2(\text{height})^2=\text{hypotenuse}^2$

$\text{height}^2=\frac{\text{hypotenuse}^2}{2}$

$\text{height}=\sqrt{\frac{\text{hypotenuse}^2}{2}}$

Simplify.

$\text{height}=\frac{\text{hypotenuse}}{\sqrt2}$

Multiply the fraction by one in the form of:

$1\rightarrow \frac{\sqrt{2}}{\sqrt{2}}$

Substitute.

$\text{height}=\frac{\text{hypotenuse}}{\sqrt2}\times \frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$\text{height}=\frac{\text{hypotenuse}(\sqrt2)}{2}{}$

Now, substitute in the value of the hypotenuse to find the height for the given triangle.

$\text{height}=\frac{144\sqrt2}{2}$

Simplify.

$\text{height}=72\sqrt2$

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Example Question #12 : How To Find The Height Of A 45/45/90 Right Isosceles Triangle

If the hypotenuse of an isoceles right triangle is , what is the length of the height of the triangle?

Possible Answers:

$95\sqrt2$

$\frac{95\sqrt2}{2}$

$95\sqrt3$

Correct answer:

$\frac{95\sqrt2}{2}$

Explanation:

An isoceles right triangle is another way of saying that the triangle is a 45-45-90 triangle.

Now, recall the Pythagorean theorem:

$\text{base}^2+\text{height}^2=\text{hypotenuse}^2$

Because we are working with a 45-45-90 triangle, the base and the height have the same length. We can rewrite the above equation as the following:

$\text{height}^2+\text{height}^2=\text{hypotenuse}^2$

$2(\text{height})^2=\text{hypotenuse}^2$

$\text{height}^2=\frac{\text{hypotenuse}^2}{2}$

$\text{height}=\sqrt{\frac{\text{hypotenuse}^2}{2}}$

Simplify.

$\text{height}=\frac{\text{hypotenuse}}{\sqrt2}$

Multiply the fraction by one in the form of:

$1\rightarrow \frac{\sqrt{2}}{\sqrt{2}}$

Substitute.

$\text{height}=\frac{\text{hypotenuse}}{\sqrt2}\times \frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$\text{height}=\frac{\text{hypotenuse}(\sqrt2)}{2}{}$

Now, substitute in the value of the hypotenuse to find the height for the given triangle.

$\text{height}=\frac{95\sqrt2}{2}$

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Example Question #73 : 45/45/90 Right Isosceles Triangles

If the hypotenuse of an isoceles right triangle is , what is the length of the height?

Possible Answers:

$13\sqrt3$

$26\sqrt2$

$16\sqrt3$

Correct answer:

$26\sqrt2$

Explanation:

An isoceles right triangle is another way of saying that the triangle is a 45-45-90 triangle.

Now, recall the Pythagorean theorem:

$\text{base}^2+\text{height}^2=\text{hypotenuse}^2$

Because we are working with a 45-45-90 triangle, the base and the height have the same length. We can rewrite the above equation as the following:

$\text{height}^2+\text{height}^2=\text{hypotenuse}^2$

$2(\text{height})^2=\text{hypotenuse}^2$

$\text{height}^2=\frac{\text{hypotenuse}^2}{2}$

$\text{height}=\sqrt{\frac{\text{hypotenuse}^2}{2}}$

Simplify.

$\text{height}=\frac{\text{hypotenuse}}{\sqrt2}$

Multiply the fraction by one in the form of:

$1\rightarrow \frac{\sqrt{2}}{\sqrt{2}}$

Substitute.

$\text{height}=\frac{\text{hypotenuse}}{\sqrt2}\times \frac{\sqrt{2}}{\sqrt{2}}$

Solve.

$\text{height}=\frac{\text{hypotenuse}(\sqrt2)}{2}{}$

Now, substitute in the value of the hypotenuse to find the height for the given triangle.

$\text{height}=\frac{52\sqrt2}{2}$

Simplify.

$\text{height}=26\sqrt2$

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Example Question #13 : How To Find The Height Of A 45/45/90 Right Isosceles Triangle

Find the height of this triangle:

Isosceles right

Possible Answers:

12.25

4.77

1.87

Correct answer:

4.77

Explanation:

To find the height, use the Pythagorean Theorem. One of the legs is the missing side, and the other is 1.5, half of 3. The hypotenuse is 5:

(1.5)^2 + x^2 = 5^ 2

2.25 + x^2 = 25

x^2 = 22.75

$x \approx 4.77$

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

$What\;is\;the\;area\;of\;\Delta ABC?$

Possible Answers:

4.5

6.5

Correct answer:

4.5

Explanation:

$Area\;\Delta=\frac{1}{2}(base)(height)$

$\frac{1}{2}(3)(3)=\frac{9}{2}=4.5$

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

Consider an isosceles triangle with a height of 24 and a base of 12. What is the area of this triangle?

Isoc._triangle

Possible Answers:

121

169

155

144

Correct answer:

144

Explanation:

The formula for the area of a trianlge is A = base * height * (1/2).

We're lucky here, because the question gives us all of the values we need. We simply need to plug them in:

A = base * height * (1/2) = 12 * 24 * (1/2) = 12 * 12 = 144

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Example Question #2 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

Calculate the area of an isosceles right triangle who's hypotenuse is $4\sqrt{2}$ inches.

Possible Answers:

$16 \text{ in.}^2$

$8 \text{ in.}^2$

$2 \text{ in.}^2$

$4 \text{ in.}^2$

Correct answer:

$8 \text{ in.}^2$

Explanation:

The formula for the area of a triangle, right or not, is one half the base times height.

In this case, they are both $4 \text{ in.}$ Therefore, the respective values are entered, yielding:

$\frac{1}{2}(4\text{ in.})(4\text{ in.})=8 \text{ in.}^2$

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

The side length of the 45-45-90 right triangle is $20\sqrt{2}$ , find the area of the right triangle.

Possible Answers:

200

400

$200\sqrt{2}$

$400\sqrt{2}$

$100\sqrt{2}$

Correct answer:

400

Explanation:

The area of a triangle is:

$A=\frac{b*h}{2}$ where b=base, h=height, and A=area

$A=\frac{20\sqrt{2}*20\sqrt{2}}{2}=\frac{400\sqrt{4}}{2}=200\sqrt{4}=200*2=400$

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

Find the area of the triangle below.

Possible Answers:

Correct answer:

Explanation:

The key to finding the area of our triangle is to reaize that it is isosceles and therefore is a 45-45-90 triangle; therefore, we know the legs of our triangle are congruent and that each can be found by dividing the length of the hypotenuse by $\sqrt{2}$ .

$\frac{8}{\sqrt{2}}=\frac{8}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}}=\frac{8\sqrt{2}}{2}$

Rationalizing the denominator simplifies our result; however, we are interested in the area, not just the length of a leg; we remember that the formula for the area of a triangle is

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

where is the base and is the height; however, in our right triangle, the base and height are simply the two legs; therefore, we can calculate the area by substituting.

$A=\frac{1}{2}(\frac{8\sqrt2}{2})(\frac{8\sqrt2}{2})=\frac{128}{8}=16$

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

Find the area of a 45/45/90 triangle that has a hypotenuse of $5\sqrt3$ .

Possible Answers:

$\frac{5\sqrt{6}}{2}$

$\frac{75}{4}$

$\sqrt{\frac{75}{2}}$

37.5

$\frac{75}{2}$

Correct answer:

$\frac{75}{4}$

Explanation:

To find the area of a triangle, we must use $Area=\frac{1}{2}b\cdot h$ where b=base and h=height.

In the problem, the only information given is what type the triangle is and what its hypotenuse is.

Given the area equation, the problem hasn't given any numbers that can be substituted into the equation to solve for an area. This means that the hypotenuse value must be used to determine the height and the base.

Because this is a 45/45/90 triangle, this means that it is also isosceles. Therefore, we can logic out that the base and the height must be the same.

The missing sides can be calulated in one of two ways:

1. Using the Pythagorean Theorem a^2+b^2=c^2

2. Or using Find_the_leg_length_resolution

If we were to use the Pythagorean Theorem, since we've already determined that b=h, that mean a=b in the equation. Let's say that a=b=s .

That means the Pythagorean Theorem can be rewritten as:

s^2+s^2=c^2

${\color{Blue} 2s^2=c^2}$

Now to substitute in the value of c to solve for the height and base.

$2s^2=(5\sqrt{3})^2$

$2s^2=25\sqrt{9}$

2s^2=75

$s^2=\frac{75}{2}$

$\sqrt{s^2}=\sqrt{\frac{75}{2}}$

$s=\frac{5\sqrt{6}}{2}$

Now that we have the base and the height, we can substitute the values into the area equation and get the triangle's area.

$Area=\frac{1}{2}\cdot\frac{5\sqrt{6}}{2}\cdot\frac{5\sqrt{6}}{2}$

$Area=\frac{1}{2}\cdot\frac{(25\cdot 6)}{4}$

$Area = \frac{1}{2}\cdot\frac{150}{4}$

$Area = \frac{1}{2}\cdot \frac{75}{2} = {\color{Blue} \frac{75}{4}}$

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Basic Geometry : 45/45/90 Right Isosceles Triangles

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #11 : How To Find The Height Of A 45/45/90 Right Isosceles Triangle

Example Question #12 : How To Find The Height Of A 45/45/90 Right Isosceles Triangle

Example Question #73 : 45/45/90 Right Isosceles Triangles

Example Question #13 : How To Find The Height Of A 45/45/90 Right Isosceles Triangle

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

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

Example Question #2 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

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

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

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

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