General Idea:
By using the Pythagorean theorem which states that in a right triangle the sum of square of two legs is equal to the square of hypotenuse. We can find the length of the third side.
Given:
[tex] a= 4 \; in\\\\c=8\; in\\\\a^2+b^2=c^2\\\\4^2+b^2=8^2\\16+b^2=64\\Subtracting \; 16 \; on \; both \; sides\\\\b^2=64-16\\b^2=48\\Take \; square \; root \; on \; both \; sides\\\\b= \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4 \sqrt{3} inches [/tex]
We can see shortest side a is 4 inches, hypotenuse is c is 8 inches and middle side is 4√3. This is quality of a special triangle. The smallest side (a) is opposite the smallest angle (30°), the middle side (b) is opposite the angle (60°) and the longest side (c) is opposite the largest angle (90°).
The triangle 30°-60° - 90° triangle will be of the ratio 1:√3:2.
Conclusion:
The length of the other leg is 4√3 in
The measures of the acute angles are : 30° and 60°
