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:
[tex]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\\\\[/tex]
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
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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