right triangle ABC has side lengths AB=21, BC=28 and AC=35. Find the ratio of the side opposite <C to the hypotenuse of triangle ABC. Express your ratio as a decimal.
Let's label the side opposite angle C as x.
By the Pythagorean Theorem, we have:
AB² + BC² = AC²
21² + 28² = 35²
441 + 784 = 1225
1225 = 1225
So, the lengths of the sides satisfy the Pythagorean Theorem, meaning that triangle ABC is a right triangle.
Using the angle C, we can set up the cosine ratio:
cos(C) = adj/hyp
cos(C) = x/35
Solving for x, we get:
x = 35 * cos(C)
To find cos(C), we can use the cosine rule:
cos(C) = (AB² + BC² - AC²) / (2 * AB * BC)
cos(C) = (21² + 28² - 35²) / (2 * 21 * 28)
cos(C) = (441 + 784 - 1225) / (2 * 21 * 28)
cos(C) = 0 / (2 * 21 * 28)
cos(C) = 0
Therefore, x = 35 * cos(C) = 35 * 0 = 0.
The ratio of the side opposite angle C to the hypotenuse of triangle ABC is x/AC = 0/35 = 0.