In a right triangle, the hypotenuse is 8 and one of the acute angles is 45 °. Find the area of the triangle.

Given:

Right-angled triangle ABC

angle С = 90 degrees,

AB – hypotenuse,

AB = 8,

angle A = 45 degrees.

Find the area of ​​the triangle ABC, that is, S ABC -?

Decision:

1. Consider a right-angled triangle ABC. The sum of the degree measures of the angles of a triangle is 180 degrees. Then angle B = 180 – angle A – angle C;

angle B = 180 – 45 – 90;

angle B = 45 degrees.

Therefore, the right-angled triangle ABC is also isosceles, then AC = BC.

2. By the Pythagorean theorem (the square of the hypotenuse is equal to the sum of the squares of the legs):

AC ^ 2 + BC ^ 2 = AB ^ 2 (let AB = BC = x centimeters);

x ^ 2 + x ^ 2 = 8 ^ 2;

2 * x ^ 2 = 64;

x ^ 2 = 64: 2;

x ^ 2 = 32.

3. S ABC = 1/2 * AC * BC;

S ABC = 1/2 * 32;

S ABC = 16.

Answer: 16.