Question by Photo:

Answer:

1.  Length of AB:  This can be found by using the Pythagorean Theorem.  In our case,

AB^2 + AC^2 = BC^2

We know AC = 3sqrt(5) and BC = 4sqrt(5).  Plug those numbers in to solve for AB like so:

AB^2 + [3sqrt(5)]^2 = [4sqrt(5)]^2

Now, solve (the “squared” must distribute to the 3 and the sqrt(5) on the left side, and same on the right side):

AB^2 + 9*5 = 16*5

AB^2 + 45 = 80

AB^2 = 35

AB = sqrt(35)  Final answer.

2.  Find the area of triangle ABC

Use AC as the base and AB as the height.  Use the formula:  Area = 1/2 * base * height

Area = 1/2 * AC * AB

Area = 1/2 * 3sqrt(5) * sqrt(35)

Area = 3/2 * sqrt(175)  [you can multiply the 1/2 and the 3… And you can multiply the 5 by 35 inside of the square roots]

Now, you probably need to simplify the radical:

Area = 3/2 * sqrt(175) = 3/2 *sqrt(25*7) = 3/2 sqrt(25)*sqrt(7) = 3/2 * 5 * sqrt(7) = 15/2 * sqrt(7)