Question

Triangle A B C is shown. Angle A C B is a right angle, angle C A B is 60 degrees, and angle A B C is 30 degrees. The length of the hypotenuse is 10.

What are the lengths of the other two sides of the triangle?

AC = 5 and BC = 5
AC = 5 and BC = 5 StartRoot 5 EndRoot
AC = 5 StartRoot 3 EndRoot and BC = 5
AC = 5 and BC = 5 StartRoot 3 EndRoot

Answer 1

AC = 5 and BC = 5 StartRoot 3 EndRoot.

explaination..

In the given triangle ABC, angle ACB is a right angle, so by the Pythagorean theorem, we can find the length of the hypotenuse AB as follows:

AB^2 = AC^2 + BC^2

Since AB is given to be 10, we can substitute this value and solve for the other two sides as:

10^2 = AC^2 + BC^2

100 = AC^2 + BC^2

Now, we are given that angle CAB is 60 degrees and angle ABC is 30 degrees. Since the three angles of a triangle add up to 180 degrees, we can find angle BAC as:

angle BAC = 180 - angle CAB - angle ABC

angle BAC = 180 - 60 - 30

angle BAC = 90 degrees

Thus, triangle ABC is a right triangle with hypotenuse AB and angle BAC as 90 degrees. Since angle BAC is 90 degrees, AC and BC are the two legs of the triangle.

We can solve the equation 100 = AC^2 + BC^2 by substitution. We know that AC = BC because the triangle is an equilateral triangle, so we can substitute AC for BC:

100 = AC^2 + AC^2

100 = 2AC^2

AC^2 = 50

AC = 5 StartRoot 2 EndRoot

Since AC = BC, we have:

AC = 5 StartRoot 2 EndRoot and BC = 5 StartRoot 2 EndRoot

Therefore, the lengths of the other two sides of the triangle are AC = 5 StartRoot 2 EndRoot and BC = 5 StartRoot 2 EndRoot.