Question

The perimeter of a rectangle is 42, and its diagonal is 15. Find its length and width.
length (larger side) _____
width (smaller side) _____

Answer 1

2a+2b=42
a^2+b^2=15^2

Answer 2

a^2+b^2=225
2(a+b)=42
(a+b)=21

Answer 3

b=21-a
plug that into
a^2+b^2=225
a^2+(21-a)^2=225

Answer 4

The diagonal makes the rectangle into two right triangles. The hypo for each triangle is 15. Lets say that the short side is a and the long side is b.
Perimeter= a+a+b+b=42
Pythagorean= a^2 + b^2= 15^2

Simplify perimeter 1st
2a+2b=42
a+b=21
Then simplify the Pythagorean one
a^2 + b^2=225

It's easiest to use substitution in this one: make the first one in terms of b (or a), then plug that equation in wherever you see a b in the second equation.
a+b =21
b= 21-a

a^2+ b^2 = 225
a^2+ (21-a)^2=225

Then solve for a.
a^2+ (21-a)(21-a)=225
a^2+ 441 -21a -21a +a^2 =225
2a^2+ 441 -42a =225
2a^2-42a+216=0

Factor that, and when you find out what a is, plug the value into its rightful place in one of the original equations and solve for b.