Assume that the wooden triangle shown is a right triangle.
a. write an equation using the pythagorean theorem and the measurements provided in the diagram.
hint: (leg 1)2 + (leg 2)2 = (hypotenuse)2
b. transform each side of the equation to determine if it is an identity.

a.

a^2 + b^2 = c^2

The legs are a and b. c is the hypotenuse.

Let a = 6x + 9y; b = 8x + 12y; c = 10x + 15y

The equation is:

(6x + 9y)^2 + (8x + 12y)^2 = (10x + 15y)^2

b.

Now we square each binomial and combine like terms on each side.

36x^2 + 108xy + 81y^2 + 64x^2 + 192 xy + 144y = 100x^2 + 300xy + 225y^2

36x^2 + 64x^2 + 108xy + 192xy + 81y^2 + 144y^2 = 100x^2 + 300xy + 225y^2

100x^2 + 300xy + 225y^2 = 100x^2 + 300xy + 225y^2

The two sides are equal, so it is an identity.

see below

Step-by-step explanation:

a  The pythagorean theorem is a^2 + b^2 = c^2  where a and b are the legs and c is the hypotenuse

a^2 + b^2 = c^2

(6x+9y)^2 + (8x+12y)^2 = (10x + 15y)^2

b solve

(6x+9y)^2 + (8x+12y)^2 = (10x + 15y)^2

(6x+9y)(6x+9y) + (8x+12y)(8x+12y) = (10x + 15y)(10x+15y)

Factor out the common factors

3(2x+3y)3(2x+3y) + 4(2x+3y)4(2x+3y) = 5(2x+3y)5(2x+3y)

Rearrange

9 (2x+3y)^2 +16 (2x+3y)^2 = 25(2x+3y)^2

Divide each side by(2x+3y)^2

9 (2x+3y)^2/ (2x+3y)^2 +16 (2x+3y)^2/(2x+3y)^2 = 25(2x+3y)^2/(2x+3y)^2

9 + 16 = 25

25=25

This is true, so it is an identity