In ΔABC, m∠ACB = 90°, and m∠ACD = 45°.

Find AC, if CD = 6 2 in.

How do I know the lengths of the sides if I only know the altitude?

/AC/=/AD/=x

/CD/=6
Using pyth.
/CD/^2=/AC/^2+/AD/^2
6^2=X^2+X^2
36=2X^2
X^2=18
X=√18=4.242cm
:./AC/=4.242cm

thanks

12 inches

To find the length of side AC in a right triangle, you can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

In this case, AC is the hypotenuse, and CD is one of the other two sides. We also know that angle ACD is 45 degrees, which means that triangle ACD is a 45-45-90 triangle. In a 45-45-90 triangle, the two legs (CD and AD) are congruent, and the length of the hypotenuse (AC) is equal to the length of a leg times the square root of 2.

Given that CD = 6 inches, we can find AC as follows:

1. Calculate the length of AD by using the fact that in an isosceles right triangle, the legs are congruent. Since CD = AD, we know that AD = 6 inches.

2. Use the Pythagorean theorem to find the length of AC. We have AC² = AD² + CD². Plugging in the known values, we get AC² = 6² + 6² = 36 + 36 = 72.

3. Take the square root of both sides to find AC. Thus, AC = sqrt(72) which is approximately 8.49 inches.

Therefore, the length of side AC is approximately 8.49 inches.

Since you have bisected the 90° angle,

∠A = 45° and ∠B = 45°
so you have isosceles triangles, and AC = CD = your given