In a right-angled triangle, the acute angle is 45 degrees, and the hypotenuse is 3√2 cm.

In a right-angled triangle, the acute angle is 45 degrees, and the hypotenuse is 3√2 cm. Find the leg and area of this triangle.

1. It is known that

a) the sum of the angles of the triangle is 180 degrees,

b) the area S of a right-angled triangle is equal to half the product of its legs,

c) the cos of the angle is the ratio of the adjacent leg to the hypotenuse.

2. According to the condition of the problem, one acute angle A of the right-angled triangle ABC is equal to 45 degrees. Find the second acute angle C.

Angle C = 180 degrees – 90 degrees – 45 degrees = 45 degrees.

So our triangle is isosceles, and the height of the isosceles triangle, lowered from the top B is also the median, that is, it divides the base of the AC at point O in half.

3. We calculate the leg of the triangle AB if it is known that

AO = 1/2 AC = 3 * 2 ^ 1/2: 2.

cosA = cos45 degrees = 2 ^ 1/2: 2.

cosA = AO: AB where we find

AB = AO: cosA = 3 * 2 ^ 1/2: 2: (2 ^ 1/2: 2) = 3cm.

4. Calculate the area of ​​the triangle S.

S = 1/2 AB ^ 2 = 3 * 3: 2 = 4.5 cm ^ 2.

Answer: The leg is 3 centimeters, the area is 4.5 square centimeters.