SOLUTION: the diagonal of a rectangle is 15cm, and the perimeter is 38cm. what is the area?

Question 1150827: the diagonal of a rectangle is 15cm, and the perimeter is 38cm. what is the area?
Found 4 solutions by Alan3354, MathLover1, ikleyn, MathTherapy:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
the diagonal of a rectangle is 15cm, and the perimeter is 38cm. what is the area?
--------
Side lengths = a & b
a^2 + b^2 = 15^2
a + b = 38/2 = 19
=================
(a+b)^2 = a^2 + 2ab + b^2 = 361
..........a^2 + b^2 = 225
------------------------------- Subtract
2ab = 136
a*b = 68
Sub for b
a*(19 - a) = 68
a^2 - 19a + 68 = 0
Solve for a
etc

Answer by MathLover1(20849)   (Show Source): You can put this solution on YOUR website!
the diagonal of a rectangle is ,
recall:
where and are sides of the rectangle
so,
..........eq.1

and, if the perimeter is , means

.......solve for
..........eq.2
go to
..........eq.1...substitute
.........solve for

...swap the sides

....both sides divide by
.......use quadratic formula

.........since we are looking for side length, we need only positive root
-> exact solution
-> approximately

now find

..........eq.2

-> exact solution
-> approximately

then, the area will be:
......using exact solutions we have

or, using approximate solutions

.....rounded it is

Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
the diagonal of a rectangle is 15cm, and the perimeter is 38cm. what is the area?
~~~~~~~~~~~~~~~~~~~~

Let "a" and "b" be the lengths of two adjacent sides of the rectangle. Then a + b = 19 = (1) + = = 225 (2) Square both sides of the equation (1); keep equation (2) as is. = = 361 (1') + = 225 (2') Subtract equation (2') from equation (1') 2ab = 361 - 225 = 136. Now, 2ab is two times the area of the rectangle; hence, the area of the rectangle is = 68 square centimeters. ANSWER. The area of the rectangle is 68 square centimeters.

Solved.

------------------

The lesson to learn:

In order to solve this problem, you do not need solve quadratic equation.

Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!
the diagonal of a rectangle is 15cm, and the perimeter is 38cm. what is the area?

I agree with @IKLEYN. Why spend a "day" solving a problem when it can take just 2 minutes? And, almost no-one who
I've ever dealt with, wants to use the quadratic equation or "Completing the Square" to solve a quadratic equation.
Let length and width be L and W, respectively
With perimeter being 38 cm, we get: 2(L + W) = 2(19)____L + W = 19 ---- eq (i)
With diagonal being 15 cm, we also get: ----- eq (ii)
------ Squaring eq (i) ------ eq (iii)
------- Substituting

We then get:
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