Question

The hypotenuse of a 45°-45°-90° triangle measures 18 cm. What is the length of one leg of the triangle?

Answer 1

The length of the one leg of the triangle is $\boxed{\bf 9\sqrt{2}\text{\bf\ cm}}$.

Further explanation:

Given:

The length of hypotenuse of a triangle is $18\text{ cm}$.

Calculation:

If one angle of the triangle is $90^{\circ}$ then the triangle is said to be a right angle triangle.

From attached Figure 1 it can be seen that $AC$ is the hypotenuse, $BC$ is the base side and $AB$ is perpendicular in a triangle $\triangle\text{ ABC}$.

In $\triangle\text{ ABC}$, the base length can be calculated as follows:

$\text{cos}45^{\circ}=\dfrac{BC}{AC}$    …… (1)

It is given that the value of $AC$ is $18\text{ cm}$. The value of   $\text{cos}45^{\circ}$ is $\frac{1}{\sqrt{2}}$.

Now, substitute $AC=18$ and $\text{cos}45^{\circ}=\frac{1}{\sqrt{2}}$ in equation (1) to obtain the value of length of $BC$.

$\begin{aligned}\dfrac{1}{\sqrt{2}}&=\dfrac{BC}{18}\\BC&=\dfrac{18}{\sqrt{2}}\\&=\dfrac{18}{\sqrt{2}}\cdot \dfrac{\sqrt{2}}{\sqrt{2}}\\&=\dfrac{18\sqrt{2}}{2}\\&=9\sqrt{2}\end{aligned}$

 

Therefore, the length of perpendicular side $BC$ is $9\sqrt{2}\text{ cm}$.

In $\triangle\text{ ABC}$, the perpendicular length can be calculated as,

$\text{sin}45^{\circ}=\dfrac{AB}{AC}$    …… (2)

Now, substitute $AC=18$ and $\text{sin}45^{\circ}=\frac{1}{\sqrt{2}}$ in equation (2) to obtain the length of side $AB$ as follows:

$\begin{aligned}\dfrac{1}{\sqrt{2}}&=\dfrac{AB}{18}\\AB&=\dfrac{18}{\sqrt{2}}\\&=\dfrac{18}{\sqrt{2}}\cdot \dfrac{\sqrt{2}}{\sqrt{2}}\\&=\dfrac{18\sqrt{2}}{2}\\&=9\sqrt{2}\end{aligned}$  

Therefore, the length of perpendicular side $AB$ is $\boxed{\bf 9\sqrt{2}\text{\bf cm}}$.

Learn more:

1. Learn more about graph: brainly.com/question/2334270

2. Learn more about problem on triangle brainly.com/question/7437053

3. Learn more about problem on angle brainly.com/question/3717797

Answer details:

Grade: Middle school

Subject: Mathematics

Chapter: Triangles

Keywords:  Triangle, perpendicular, base, hypotenuse, cos, sin, cos45, sin45, right angles triangle, figure, length, 45-45-90, 18cm, 9square root 2.

Answer 2

Special prpoerty of 45-45-90 triangle
if the legs are x then the hyptonuse is x√2

18=hyptonuse=x√2

18=x√2
divide both sides by √2
18/√2
make denomenator nice
times by √2/√2
(18√2)/(2)
9√2

one leg is 9√2 cm