Question

The diagonal of a rectangle is 15 cm, and the perimeter is 38 cm. What is the area?

Answer 1

Answer:  68 cm²

Step-by-step explanation:

Area = length × width

Let the width be represented by w and the length be represented by L.

Pythagorean Theorem: w² + L² = diagonal²   ⇒   w² + L² = 15²

Perimeter = 2w + 2L  ⇒   38 = 2w + 2L  

19 = w + L      divided both sides by 2

19 - w = L      subtracted w from both sides

Substitute L with 19 - w in the Pythagorean Theorem equation:

w² + (19 - w)² = 15²

w² + 361 - 38w +w² = 15²

2w² - 38w + 361 = 225

2w² - 38w + 136 = 0

w² - 19w + 68 = 0       divided both sides by 2

Use Quadratic Formula to solve for w:

$x=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}\\\\w=\dfrac{19\pm \sqrt{(-19)^2-4(1)(68)}}{2}\\\\w=\dfrac{19\pm \sqrt{361-272}}{2}\\\\w=\dfrac{19\pm \sqrt{89}}{2}\\\\w=\dfrac{19\pm 9.434}{2}\\\\w=\dfrac{28.434}{2}\quad or\quad \dfrac{9.566}{2}\\\\w=14.217\quad or\quad 4.783$

Use the Perimeter equation to solve for L:

19 - w = L                     19 - w = L

19 - 14.217 = L              19 - 4.783 = L

4.783 = L                      14.217 = L

Notice that we end up with the same dimensions regardless of which value we use for w

Now, let's find the Area:

A = L × w

  = 14.217 × 4.783

  = 68

Answer 2

Answer:

a² + b² = 15² = 225

P = 2a + 2b = 2(a + b) = 38

P² = 4(a + b)² = 38²

4(a² + 2ab + b²) = 1444

Substitute a² + b² with 225

4(2ab + 225) = 1444

2ab + 225 = 361

2ab = 136

ab = 68 cm²