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Question

An architect wants to draw a rectangle with a diagonal of 25 millimeters. The length of the rectangle is to be 10 millimeters more than twice the width. What dimensions should she make the rectangle? 

Answer 1

The diagonal is acting as the hypotenuse of a right triangle. The length and width are the legs.

Use the Pythagorean Theorem c²=a²+b².or, in this case d²=l²+w².

d (diagonal) = 25

w (width) = w

l (length) = 2w+10

25²=(2w+10)²+w²

25²=625

Using the FOIL method, expand (2w+10)²=(2w+10)(2w+10)=4w²+40w+100

625=4w²+40w+100+w²

Combine like terms and set the problem equal to zero.

0=5w²+40w-525

Divide all terms by 5.

0=w²+8w-105

Factor into two binomials.(I'd be glad to explain this process)

0=(w+15)(w-7)

Set each binomial equal to zero.

w+15=0 w-7=0

w=-15 w=7 (Discard the negative answer since you are looking for a distance.)

w=7mm

Now substitute width in l=2w+10

l=2w+10

l=2(7)+10

l=14+10

l=24 mm

Answer 2

d^2 = l^2 + w^2 = (10+2w)^2 + w^2 = 100+40w + 4w^2 + w^2

25^2= 5w^2 + 40w + 100

5w^2 + 40w - 525 = 0

w^2 +8w - 105 = 0

w = -8/2 + or - (1/2)sqr(64+420) = -4 + or - 11= -15 or 7 wide

l= 10 + 2(7) = 24 long

check the solution

24^2 + 7^2 = 576 + 49 = 625 = 25^2

Answer 3

We're going to draw a picture of a rectangle and a line right up the diagonal, making a right triangle. And we'll use the pythagorean theorem to aid us. Let L = length and W = width. By the information given: L = 2W + 10, 2 times the width plus 10. Based on the triangle you have with this picture you will have L² + W² = 25², according to the pythagorean theorem. But we also know what L is in terms of W. So,

(2W+10)² + W² = 25².

Now simplify, FOIL everything out, move everything to one side and factor. You will most likely get a positive and negative answer for the width, which is a distance, so discard the negative answer. Since you have an answer for W, now plug that into L = 2W + 10 to get the length.