32)
Since the sum of the angle measures in every triangle is 180°,
and we know that two of the angles are 90° and 45°.....
the measure of the third angle = m∠A = 180° - 90° - 45° = 45°
Since m∠A = m∠C , triangle ABC is isosceles and AB = BC
By the Pythagorean Theorem...
(AB)2 + (BC)2 = n2
We can replace AB with BC since they are the same length.
(BC)2 + (BC)2 = n2
Combine like terms.
2(BC)2 = n2
Divide both sides of the equation by 2 .
(BC)2 = n2 / 2
Take the positive square root of both sides.
BC = \(\sqrt{\frac{n^2}{2}}\)
Simplify.
BC = \(\frac{\sqrt{n^2}}{\sqrt2}\)
BC = \(\frac{n}{\sqrt2}\)
Rationalize denominator.
BC = \(\frac{n\sqrt2}{2}\)
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