Given a triangle, with the following dimensions below
[tex]\begin{gathered} \text{Opposite}=10\text{ units} \\ \text{Adjacent}=x\text{ units} \\ \text{Hypotenuse}=y\text{ units} \\ \theta=45^o \end{gathered}[/tex]To find the value of x, we use SOHCAHTOA,
Where
[tex]\tan \theta=\frac{Opposite}{Adjacent}[/tex]Substitute the values into the formula above
[tex]\begin{gathered} \tan \theta=\frac{Opposite}{Adjacent} \\ \tan 45^o=\frac{10}{x} \\ \text{Where }\tan 45^o=1 \\ 1=\frac{10}{x} \\ \text{Crossmultiply} \\ x=10\text{ units} \end{gathered}[/tex]Thus, x = 10 units
To find the value of y, using the Pythagorean theorem
The Pythagorean theorem is
[tex](\text{HYP)}^2=(OPP)^2+(\text{ADJ)}^2[/tex]Substitute the values to find the value of y
[tex]\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\times100}=\sqrt[]{100}\times\sqrt[]{2} \\ y=10\times\sqrt[]{2} \\ y=10\sqrt[]{2}\text{ units} \end{gathered}[/tex]Hence, the values of x and y are
[tex]\begin{gathered} x=10\text{ units} \\ y=10\sqrt[]{2}\text{ units} \end{gathered}[/tex]