In a right-angled triangle ABC, angle C = 90 degrees AB = 4 angle ABC = 45 degrees.

| July 9, 2021 | education

In a right-angled triangle ABC, angle C = 90 degrees AB = 4 angle ABC = 45 degrees. Find the AC and the height of the CD drawn to the hypotenuse.

The sum of the angles of the triangle is 180 degrees, so the angle BAC = 180-angle C-angle ABC = 180-90-45 = 45 degrees. The cosine of the angle BAC is the ratio of the leg AC to the hypotenuse AB: cosBAC = AC / AB, AC = AB * cosBAC = AB * cos45 = 4 * √2 / 2 = 2√2.
Consider triangle ADC. CD – height, angle D – straight line, angle DAC – 45 degrees, so the angle ACD is also 45 degrees. Therefore, triangle ADC is isosceles with base AC, sides AD and CD are equal to each other. AD – half of AB, because in an isosceles triangle ABC, the height of CD is the median and divides the hypotenuse of AB in half. CD = AD = AB / 2 = 4/2 = 2.



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