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

The area of the rectangle is 68.16 cm²

A rectangle is a quadrilateral in which opposite sides are parallel and equal. Also each angle is a rectangle measures 90°.

Since the diagonal is 15 cm, hence let x represent the length of the rectangle and y represent the width of the rectangle. Hence:

x² + y² = 15²

x² + y² = 225   (1)

The perimeter of the rectangle is 38 cm, hence:

Perimeter = 2(x + y)

38 = 2(x + y)

x + y = 19    

y = 19 - x    (2)

x² + (19 - x)² = 225

x² + 361 - 38x + x² = 225

2x² - 38x + 136 = 0

x = 14.2 and y = 4.8

Area of the rectangle = xy = 14.2 * 4.8 = 68.16 cm²

The area of the rectangle is 68.16 cm²

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 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:

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