We will have the following:
a. We can see that the angle opposite to x is 45° and the angle opposite to 10 is also 45°; from theorems [Congruent angles that oppose sides in a triangle have that those sides are also congruent]; so:
x = 10.
Now, we determine y using the law of sines:
[tex]\frac{y}{sin(90)}=\frac{10}{sin(45)}\Rightarrow y=10\sqrt{2}[/tex]
Now, we determine the height of the triangle:
[tex]\begin{gathered} 10^2=(5\sqrt{2})^2+h^2\Rightarrow h^2=100-50 \\ \\ \Rightarrow h=5\sqrt{2} \end{gathered}[/tex]
So, the perimeter is:
[tex]P=10+10+10\sqrt{2}\Rightarrow P=20+10\sqrt{2}[/tex]
The area is:
[tex]A=\frac{(10\sqrt{2})(5\sqrt{2})}{2}\Rightarrow A=50[/tex]
b. Same as in the previous triangle we will determine x and y:
x = 15
[tex]\frac{y}{sin(60)}=\frac{15}{sin(60)}\Rightarrow y=15[/tex]
Now, we determine the height of the triangle:
[tex]h=\sqrt{(15)^2-(\frac{15}{2})^2}\Rightarrow h=\frac{15\sqrt{3}}{2}[/tex]
So, the perimeter is:
[tex]P=15+15+15\Rightarrow P=45[/tex]
The area is:
[tex]A=\frac{(15)(\frac{15\sqrt{3}}{2})}{2}\Rightarrow A=\frac{225\sqrt{3}}{4}[/tex]
