Question

find k.
triangle with a right angle, two 45° angles, one leg 3 mm, the other leg k
write your answer in simplest radical form.
blank millimeters

Explanation:

Step1: Identify triangle type

This is a right isosceles triangle (two 45° angles, right angle), so the two legs are equal, and we can use trigonometry or Pythagoras. We use sine: $\sin(45^\circ)=\frac{\text{opposite}}{\text{hypotenuse}}$

Step2: Substitute known values

$\sin(45^\circ)=\frac{3}{k}$, and $\sin(45^\circ)=\frac{\sqrt{2}}{2}$

Step3: Solve for k

Rearrange: $k=\frac{3}{\sin(45^\circ)}=\frac{3}{\frac{\sqrt{2}}{2}}$
Simplify: $k=3\times\frac{2}{\sqrt{2}}=\frac{6}{\sqrt{2}}=3\sqrt{2}$

Answer:

$3\sqrt{2}$ millimeters