Question

Use special right triangle patterns for the problem. Do not enter decimals. use "sqrt" for square roots.

Answer 1

Given a triangle, with the following dimensions below

`\begin{gathered} \text{Opposite}=10\text{ units} \\ \text{Adjacent}=x\text{ units} \\ \text{Hypotenuse}=y\text{ units} \\ \theta=45^o \end{gathered}`

To find the value of x, we use SOHCAHTOA,

Where

`\tan \theta=(Opposite)/(Adjacent)`

Substitute the values into the formula above

`\begin{gathered} \tan \theta=(Opposite)/(Adjacent) \\ \tan 45^o=(10)/(x) \\ \text{Where }\tan 45^o=1 \\ 1=(10)/(x) \\ \text{Crossmultiply} \\ x=10\text{ units} \end{gathered}`

Thus, x = 10 units

To find the value of y, using the Pythagorean theorem

The Pythagorean theorem is

`(\text{HYP)}^2=(OPP)^2+(\text{ADJ)}^2`

Substitute the values to find the value of y

`\begin{gathered} y^2=10^2+x^2 \\ \text{Where x}=10 \\ y^2=10^2+10^2 \\ y^2=100+100=200 \\ y^2=200 \\ \text{Square of both sides} \\ \sqrt[]{y^2}=\sqrt[]{200} \\ y=\sqrt[]{2*100}=\sqrt[]{100}*\sqrt[]{2} \\ y=10*\sqrt[]{2} \\ y=10\sqrt[]{2}\text{ units} \end{gathered}`

Hence, the values of x and y are

`\begin{gathered} x=10\text{ units} \\ y=10\sqrt[]{2}\text{ units} \end{gathered}`