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Basic Geometry : 45/45/90 Right Isosceles Triangles

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Example Questions

Example Question #1091 : Basic Geometry

Find the area of the triangle if the radius of the circle is .

Possible Answers:

$64\pi$

128

256

Correct answer:

Explanation:

The two tick marks on the image indicate that those sides are congruent; therefore, this is an isosceles right triangle.

Notice that the hypotenuse of the triangle is also the diameter of the circle. Recall the relationship between the diameter of a circle and its radius:

$\text{Hypotenuse}=\text{Diameter}=2\times\text{radius}$

Substitute in the value of the radius to find the length of the diameter.

$\text{Hypotenuse}=2\times 8$

Simplify.

$\text{Hypotenuse}=16$

Now, use the Pythagorean theorem to find the lengths of the missing sides of the triangle.

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

$2(\text{side})^2=\text{Hypotenuse}^2$

$\text{side}^2=\frac{\text{Hypotenuse}^2}{2}$

Now, recall how to find the area of a triangle:

$\text{Area}=\frac{\text{base}\times\text{height}}{2}$

In this case, because we have an isosceles right triangle,

$\text{base}=\text{height}=\text{side}$

$\text{Area}=\frac{\text{side}^2}{2}$

Take the equation derived from the Pythagorean theorem and plug it in to the equation above.

$\text{Area}=\frac{\text{Hypotenuse}^2}{2}\times \frac{1}{2}$

Simplify.

$\text{Area}=\frac{\text{Hypotenuse}^2}{4}$

Now, substitute in the value of the hypotenuse to find the area of the triangle.

$\text{Area}=\frac{16^2}{4}$

Solve.

$\text{Area}=64$

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Example Question #1092 : Basic Geometry

Find the area of the triangle if the radius of the circle is $\frac{1}{2}$ .

Possible Answers:

$\frac{1}{4}$

$\frac{1}{8}$

$\frac{1}{16}$

$\frac{1}{2}$

Correct answer:

$\frac{1}{4}$

Explanation:

The two tick marks on the image indicate that those sides are congruent; therefore, this is an isosceles right triangle.

Notice that the hypotenuse of the triangle is also the diameter of the circle. Recall the relationship between the diameter of a circle and its radius:

$\text{Hypotenuse}=\text{Diameter}=2\times\text{radius}$

Substitute in the value of the radius to find the length of the diameter.

$\text{Hypotenuse}=2\times \frac{1}{2}$

Simplify.

$\text{Hypotenuse}=1$

Now, use the Pythagorean theorem to find the lengths of the missing sides of the triangle.

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

$2(\text{side})^2=\text{Hypotenuse}^2$

$\text{side}^2=\frac{\text{Hypotenuse}^2}{2}$

Now, recall how to find the area of a triangle:

$\text{Area}=\frac{\text{base}\times\text{height}}{2}$

In this case, because we have an isosceles right triangle,

$\text{base}=\text{height}=\text{side}$

$\text{Area}=\frac{\text{side}^2}{2}$

Take the equation derived from the Pythagorean theorem and plug it in to the equation above.

$\text{Area}=\frac{\text{Hypotenuse}^2}{2}\times \frac{1}{2}$

Simplify.

$\text{Area}=\frac{\text{Hypotenuse}^2}{4}$

Now, substitute in the value of the hypotenuse to find the area of the triangle.

$\text{Area}=\frac{1^2}{4}$

Solve.

$\text{Area}=\frac{1}{4}$

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Example Question #1093 : Basic Geometry

Find the area of the triangle if the radius of the circle is $\frac{3}{2}$ .

Possible Answers:

$\frac{9}{4}$

$\frac{9}{8}$

$\frac{9}{2}$

Correct answer:

$\frac{9}{4}$

Explanation:

The two tick marks on the image indicate that those sides are congruent; therefore, this is an isosceles right triangle.

Notice that the hypotenuse of the triangle is also the diameter of the circle. Recall the relationship between the diameter of a circle and its radius:

$\text{Hypotenuse}=\text{Diameter}=2\times\text{radius}$

Substitute in the value of the radius to find the length of the diameter.

$\text{Hypotenuse}=2\times \frac{3}{2}$

Simplify.

$\text{Hypotenuse}=3$

Now, use the Pythagorean theorem to find the lengths of the missing sides of the triangle.

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

$2(\text{side})^2=\text{Hypotenuse}^2$

$\text{side}^2=\frac{\text{Hypotenuse}^2}{2}$

Now, recall how to find the area of a triangle:

$\text{Area}=\frac{\text{base}\times\text{height}}{2}$

In this case, because we have an isosceles right triangle,

$\text{base}=\text{height}=\text{side}$

$\text{Area}=\frac{\text{side}^2}{2}$

Take the equation derived from the Pythagorean theorem and plug it in to the equation above.

$\text{Area}=\frac{\text{Hypotenuse}^2}{2}\times \frac{1}{2}$

Simplify.

$\text{Area}=\frac{\text{Hypotenuse}^2}{4}$

Now, substitute in the value of the hypotenuse to find the area of the triangle.

$\text{Area}=\frac{3^2}{4}$

Solve.

$\text{Area}=\frac{9}{4}$

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Example Question #1094 : Basic Geometry

Find the area of the triangle if the radius of the circle is $\frac{9}{2}$ .

Possible Answers:

$\frac{81}{2}$

$\frac{81}{4}$

Correct answer:

$\frac{81}{4}$

Explanation:

The two tick marks on the image indicate that those sides are congruent; therefore, this is an isosceles right triangle.

Notice that the hypotenuse of the triangle is also the diameter of the circle. Recall the relationship between the diameter of a circle and its radius:

$\text{Hypotenuse}=\text{Diameter}=2\times\text{radius}$

Substitute in the value of the radius to find the length of the diameter.

$\text{Hypotenuse}=2\times \frac{9}{2}$

Simplify.

$\text{Hypotenuse}=9$

Now, use the Pythagorean theorem to find the lengths of the missing sides of the triangle.

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

$2(\text{side})^2=\text{Hypotenuse}^2$

$\text{side}^2=\frac{\text{Hypotenuse}^2}{2}$

Now, recall how to find the area of a triangle:

$\text{Area}=\frac{\text{base}\times\text{height}}{2}$

In this case, because we have an isosceles right triangle,

$\text{base}=\text{height}=\text{side}$

$\text{Area}=\frac{\text{side}^2}{2}$

Take the equation derived from the Pythagorean theorem and plug it in to the equation above.

$\text{Area}=\frac{\text{Hypotenuse}^2}{2}\times \frac{1}{2}$

Simplify.

$\text{Area}=\frac{\text{Hypotenuse}^2}{4}$

Now, substitute in the value of the hypotenuse to find the area of the triangle.

$\text{Area}=\frac{9^2}{4}$

Solve.

$\text{Area}=\frac{81}{4}$

Report an Error

Example Question #1095 : Basic Geometry

Find the area of the triangle if the radius of the circle is $\frac{11}{2}$ .

Possible Answers:

$\frac{11}{4}$

$\frac{11\sqrt2}{4}$

$\frac{121}{4}$

$\frac{121}{2}$

Correct answer:

$\frac{121}{4}$

Explanation:

The two tick marks on the image indicate that those sides are congruent; therefore, this is an isosceles right triangle.

Notice that the hypotenuse of the triangle is also the diameter of the circle. Recall the relationship between the diameter of a circle and its radius:

$\text{Hypotenuse}=\text{Diameter}=2\times\text{radius}$

Substitute in the value of the radius to find the length of the diameter.

$\text{Hypotenuse}=2\times \frac{11}{2}$

Simplify.

$\text{Hypotenuse}=11$

Now, use the Pythagorean theorem to find the lengths of the missing sides of the triangle.

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

$2(\text{side})^2=\text{Hypotenuse}^2$

$\text{side}^2=\frac{\text{Hypotenuse}^2}{2}$

Now, recall how to find the area of a triangle:

$\text{Area}=\frac{\text{base}\times\text{height}}{2}$

In this case, because we have an isosceles right triangle,

$\text{base}=\text{height}=\text{side}$

$\text{Area}=\frac{\text{side}^2}{2}$

Take the equation derived from the Pythagorean theorem and plug it in to the equation above.

$\text{Area}=\frac{\text{Hypotenuse}^2}{2}\times \frac{1}{2}$

Simplify.

$\text{Area}=\frac{\text{Hypotenuse}^2}{4}$

Now, substitute in the value of the hypotenuse to find the area of the triangle.

$\text{Area}=\frac{11^2}{4}$

Solve.

$\text{Area}=\frac{121}{4}$

Report an Error

Example Question #1091 : Plane Geometry

Find the area of this triangle:

Isosceles right

Possible Answers:

7.9

7.16

6.24

5.12

Correct answer:

7.16

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

$x \approx 4.77$

Now we can find the area using the formula

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

In this case,

$\frac{1}{2}(3)(4.77) \approx 7.16$

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Example Question #1097 : Basic Geometry

Isoscelesrighttriangle 1000

In the isosceles right triangle shown here, its hypotenuse is $3\sqrt{2}$ units long. What is its area?

Possible Answers:

4.5

7.5

Correct answer:

4.5

Explanation:

By the Pythagorean Theorem, we can deduce the lengths of both of the sides of the given isosceles right triangle, since they will be congruent according to the definition of an isosceles triangle:

a^2+b^2=c^2

$a^2+b^2=(3\sqrt{2})^2$

$2a^2=(3\sqrt{2})^2$

because a=b .

2a^2=18

a^2=9

$a=\sqrt{9}=3$ .

Now we can calculate this triangle's area according to the formula for the area of a right triangle:

$A=\frac{bh}{2}=\frac{(3)(3)}{2}=\frac{9}{2}=4.5$ .

Hence, the area of this isosceles right triangle is 4.5 square units.

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Example Question #1101 : Plane Geometry

If one of the legs of a right isosceles triangle is 12cm, what is the area of the triangle?

Possible Answers:

144 cm

$72cm^{2}$

72 cm

$36 cm^{2}$

36 cm

Correct answer:

$72cm^{2}$

Explanation:

First draw the triangle with one leg as the base. Since this is an isosceles triangle both legs measure 12cm. Label each leg with its measure of 12cm. One of the legs will act as the base, and the other leg will act as the height. In this problem we don't need to worry about the hypotenuse.

We will need to use our formula for area of a triangle

$\text{Area} = \frac{1}{2} \text{ base} \times \text{height}$

When we apply our formula we have half of twelve multiplied by twelve

$\text{Area} = \frac{1}{2} (12\times 12)$

since half of twelve is six, the next step is to multiply 12 times 6, and we get 72.

$\text{Area} = (6\times 12)=72$

Since this triangle is measured in centimeters and we multiplied to get the area, our answer will be in centimeters squared.

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Example Question #1102 : Plane Geometry

The figure below shows a triangle inscribed in a circle. The diameter of the circle is $10\textup{ meters}$ . What is the area, in square meters, of the triangle?

Possible Answers:

100

$5\sqrt{5}$

Correct answer:

Explanation:

Notice that the given triangle is a right isosceles triangle. The hypotenuse of the triangle is the same as the diameter of the circle; therefore, we can use the Pythagorean theorem to find the length of the legs of this triangle.

$\text{Hypotenuse}^2=\text{side}^2+\text{side}^2$

$2(\text{side}^2)=\text{Hypotenuse}^2$

$\text{side}^2=\frac{\text{Hypotenuse}^2}{2}$

$\text{side}=\sqrt{\frac{\text{Hypotenuse}^2}{2}}=\frac{\text{Hypotenuse}\sqrt2}{2}$

Substitute in the given hypotenuse to find the length of the leg of a triangle.

$\text{side}=\frac{10(\sqrt2)}{2}$

Simplify.

$\text{side}=5\sqrt2$

Now, recall how to find the area of a triangle.

$\text{Area}=\frac{\text{base}\times\text{height}}{2}$

Since we have a right isosceles triangle, the base and the height are the same length.

$\text{Area}=\frac{5\sqrt2\times 5\sqrt2}{2}$

Solve.

$\text{Area}=25$

Report an Error

Example Question #41 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

If the hypotenuse of an isosceles right triangle is cm, what is the area of the triangle in square centimeters?

Isosceles right

Possible Answers:

125

Correct answer:

Explanation:

Isosceles right triangles are special triangles because they possess angles of the following measures: $45^\text{o}$ , $45^\text{o}$ , and $90^\text{o}$ . These triangles are known as 45-45-90 triangles and have special characteristics. Recall the Pythagorean theorem:

$a^{2}+b^{2}=c^{2}$

In this equation, is the length of the triangle's base, is equal to its height, and is equal to the length of its hypotenuse. In an isosceles right triangle, the base and the height have the same length; therefore, is equal to , and you can rewrite the Pythagorean theorem like this:

$2 a^{2}=c^{2}$

Rearrange the equation so that is isolated on one side of the equals sign. First, simplify by dividing both sides of the equation by 2.

$\frac{2a^{2}}{2}=\frac{c^{2}}{2}$

$a^{2}=\frac{c^{2}}{2}$

Next, take the square root of both sides.

$\sqrt{a^{2}}=\sqrt{\frac{c^{2}}{2}}$

$a^{2}=\sqrt{\frac{c^{2}}{2}}$

Now, plug in the value of the hypotenuse to find the height/base of the given triangle.

$a=\frac{10\sqrt2}{2}$

$a=5\sqrt2$

Now, recall how to find the area of a triangle:

$\text{Area}=\frac{1}{2}\times\text{base}\times\text{height}$

Since the base and the height are the same length, we can find the area of the given triangle.

$\text{Area}=\frac{1}{2}(5\sqrt2 \times 5\sqrt2)$

$\text{Area}=\frac{1}{2}(25\times 2)$

Solve.

$\text{Area}=25 \text{ cm}^2$

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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 #1091 : Basic Geometry

Example Question #1092 : Basic Geometry

Example Question #1093 : Basic Geometry

Example Question #1094 : Basic Geometry

Example Question #1095 : Basic Geometry

Example Question #1091 : Plane Geometry

Example Question #1097 : Basic Geometry

Example Question #1101 : Plane Geometry

Example Question #1102 : Plane Geometry

Example Question #41 : How To Find The Area Of A 45/45/90 Right Isosceles Triangle

All Basic Geometry Resources

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