In right triangle ABC, Angle C = 90 degrees, D is a point on AB, and CD is perpendicular to AB. AB = 20 and AD = 5. Find AC

I tried formulas like soacahtoa and Pythagorean theorem but I am unable to figure this out.

nevermind! i had an epiphany

Let AD = x

The Geometric Mean Theorem states that

AD*DB = CD^2
x(20-x) = 25
x = 5(2+√3)

Since right triangles ADC and ACB are similar (Angle A is common to both),

AC/x = 20/AC
AC^2 = 20x = 100(2+√3)
AC = 10√(2+√3) = 5(√2+√6)

To find AC in right triangle ABC, we can use the Pythagorean theorem.

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (in this case, AC) is equal to the sum of the squares of the lengths of the other two sides (AB and BC).

Given that AB = 20 and AD = 5, we can find BC.

Using the Pythagorean theorem, we have:

AC^2 = AB^2 - BC^2

Substituting the given values, we have:

AC^2 = 20^2 - BC^2
AC^2 = 400 - BC^2
AC^2 = 400 - (AB - AD)^2

Since CD is perpendicular to AB, triangle ADC is also a right triangle. Therefore, the length of BC is equal to the length of CD.

Using the given values, we have:

BC = CD = AB - AD
BC = CD = 20 - 5
BC = CD = 15

Substituting BC = CD = 15 into the equation for AC^2, we have:

AC^2 = 400 - (20 - 5)^2
AC^2 = 400 - 15^2
AC^2 = 400 - 225
AC^2 = 175

Taking the square root of both sides, we find:

AC = √175

Therefore, AC is equal to √175 or approximately 13.23.

To find AC, we can use the Pythagorean theorem. In a right triangle, the sum of the squares of the lengths of the two legs (the sides that are not the hypotenuse) is equal to the square of the length of the hypotenuse.

In this case, we have right triangle ABC with a right angle at C, and AC is the hypotenuse. We are given that AB = 20 and AD = 5.

To find AC, we can use the Pythagorean theorem, which states that:

AC^2 = AB^2 + BC^2

Since we know AB = 20, we need to find the value of BC.

BC is a part of the right triangle BCD, where CD is perpendicular to AB. From triangle ACD, we can use AD as the opposite side and CD as the adjacent side to find BC using the tangent function:

tan(C) = AD / CD

tan(90°) = 5 / CD
0 = 5 / CD
CD = infinity

Since CD is perpendicular to AB, it extends infinitely in the opposite direction. Therefore, BC is also infinity.

Now, we can substitute the values into the Pythagorean theorem:

AC^2 = AB^2 + BC^2
AC^2 = 20^2 + infinity^2
AC^2 = 400 + infinity^2
AC^2 = infinity

As BC is infinity in this case, AC cannot be determined using the given information.