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Basic Geometry : Plane Geometry

Study concepts, example questions & explanations for Basic Geometry

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Example Questions

Example Question #58 : How To Find The Length Of The Hypotenuse Of A 45/45/90 Right Isosceles Triangle : Pythagorean Theorem

Right triangle 7

Refer to the above figure.

True or false: $\overline{AC}$ has length $16\sqrt{2}$ .

Possible Answers:

False

True

Correct answer:

True

Explanation:

The figure shows a right triangle. The acute angles of a right triangle have measures whose sum is $90^{\circ }$ , so

$m \angle A+ m \angle C = 90^{\circ }$

Substituting $45^{\circ }$ for $m \angle C$ :

$m \angle A+ 45^{\circ } = 90^{\circ }$

$m \angle A= 45^{\circ }$

This makes $\bigtriangleup ABC$ a 45-45-90 triangle. By the 45-45-90 Triangle Theorem, the length of hypotenuse $\overline{AC}$ is equal to that of leg $\overline{BC}$ multiplied by $\sqrt{2}$ . Therefore,

$AC = AB \cdot \sqrt{2} = 16\sqrt{2}$ .

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Example Question #1041 : Basic Geometry

$\fn_cm In\;\Delta ABC,\;\angle BAC=45^{\circ }.$

$\angle BCA = \, ?$

Possible Answers:

$\fn_cm 45^{\circ}$

$135^{\circ}$

$60^{\circ}$

$90^{\circ}$

Correct answer:

$\fn_cm 45^{\circ}$

Explanation:

$The\;degrees\;in\;a\;triangle\;sum\;to\;180^{\circ}. \; 180^{\circ}-90^{\circ}-45^{\circ}=45^{\circ}.$

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

Find the height of the 45-45-90 right triangle with a hypotenuse of $15\sqrt{6}$ .

Possible Answers:

$\sqrt{6}$

$15\sqrt{2}$

7.5

$15\sqrt{3}$

Correct answer:

$15\sqrt{3}$

Explanation:

If the hypotenuse of a 45-45-90 right triangle is $15\sqrt{6}$ then:

The height and the base of the triangle will be the same length since it is a 45-45-90 triangle (isosceles).

a^2+b^2=c^2

Since both a and b will be equal let's use a=b=x and $c=15\sqrt{6}$ :

$x^2 +x^2=(15\sqrt{6})^2$

2x^2=225*6

2x^2=1350

x^2=675

$x=\sqrt{675}$

$x=\sqrt{225*3}$

$x=\sqrt{225}\, \sqrt{3}$

$x=15\sqrt{3}$

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Example Question #1041 : Plane Geometry

What is the height of a 45/45/90 triangle if its hypotenuse is $2\sqrt7$ cm.

Find_the_height

Possible Answers:

28cm

2cm

3cm

$\sqrt{14} cm$

14 cm

Correct answer:

$\sqrt{14} cm$

Explanation:

Given that is a 45/45/90 triangle, it means that it's also isosceles. Because the hypotenuse if 2√7 cm, that means that the base and the height (the two remaining sides) will be equivalent.

The length of one of the legs can be solved for in one of two ways.

1. The Pythagorean Theorem

2. Using Find_the_leg_length_resolution

Using the Pythagorean Theorem, a^2+b^2=c^2 , we've already determined that "a" and "b" are the same number. Lets say a=b=s . This allows for the equation to be rewritten as s^2+s^2=c^2 , which may be simplified into ${\color{Purple} 2s^2=c^2}$

Because s is our unknown, we will be solving for s.

$2s^2=(2\sqrt{7})^2$

$2s^2=4\sqrt{49}$

2s^2=28

$s^2=\frac{28}{2}$

s^2=14

$\sqrt{s^2}=\sqrt{14}$

${\color{Blue} s=\sqrt{14} cm}$

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Example Question #1 : 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?

Possible Answers:

$\frac{13}{2}$

$\frac{15\sqrt2}{2}$

$\frac{13\sqrt2}{2}$

Correct answer:

$\frac{13\sqrt2}{2}$

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.

$\text{height}=\frac{13\sqrt2}{2}$

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Example Question #1 : 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:

$21\sqrt3$

$\frac{21\sqrt2}{2}$

Correct answer:

$\frac{21\sqrt2}{2}$

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.

$\text{height}=\frac{21\sqrt2}{2}$

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

If the hypotenuse of an isoceles right triangle is 212 , what is the length of the height?

Possible Answers:

$106\sqrt2$

$181\sqrt2$

$126\sqrt2$

$212\sqrt2$

Correct answer:

$106\sqrt2$

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.

$\text{height}=\frac{212\sqrt2}{2}$

Simplify.

$\text{height}=106\sqrt2$

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Example Question #2 : 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:

$6\sqrt2$

$12\sqrt2$

Correct answer:

$6\sqrt2$

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.

$\text{height}=\frac{12\sqrt2}{2}$

Simplify.

$\text{height}=6\sqrt2$

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Example Question #7 : 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?

Possible Answers:

$40\sqrt2$

$24\sqrt3$

$14\sqrt2$

$28\sqrt2$

Correct answer:

$28\sqrt2$

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.

$\text{height}=\frac{56\sqrt2}{2}$

Simplify.

$\text{height}=28\sqrt2$

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Example Question #4 : 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?

Possible Answers:

$33\sqrt2$

$66\sqrt2$

$55\sqrt2$

$44\sqrt2$

Correct answer:

$33\sqrt2$

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.

$\text{height}=\frac{66\sqrt2}{2}$

Simplify.

$\text{height}=33\sqrt2$

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Basic Geometry : Plane Geometry

Study concepts, example questions & explanations for Basic Geometry

All Basic Geometry Resources

Example Questions

Example Question #58 : How To Find The Length Of The Hypotenuse Of A 45/45/90 Right Isosceles Triangle : Pythagorean Theorem

Example Question #1041 : Basic Geometry

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

Example Question #1041 : Plane Geometry

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

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

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

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

Example Question #7 : How To Find The Height Of A 45/45/90 Right Isosceles Triangle

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

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