The diagonal of a rectangle is 25 meters long and makes an angle of 36 degrees with one side of...

Question:

The diagonal of a rectangle is 25 meters long and makes an angle of 36 degrees with one side of the rectangle. Find the area and the perimeter of the parallelogram.

The Perimeter and Area of a Rectangle:

A rectangle is a closed geometrical figure that has two pairs of parallel sides. The parallel sides in a rectangle are congruent. The area of a rectangle is the amount of space covered by the figure. It is given by {eq}A = l\times w {/eq}. The perimeter of a rectangle is the sum of its four side lengths. It is twice the sum of the length and the width {eq}P = 2(l + w) {/eq}.

The diagonal of a rectangle divides the figure into two equal right-angled triangles with the base equal to the length of the triangle and the height equal to the width of the triangle. If the diagonal makes an angle of 36 degrees with the side of the rectangle and it is 25 meters long, then:

• {eq}\sin 36 = \dfrac{w}{25} {/eq}
• {eq}w = 25\sin 36 \approx 14.7\; \rm m {/eq}
• {eq}\cos 36 = \dfrac{l}{25} {/eq}
• {eq}l = 25\cos 36 \approx 20.23\; \rm m {/eq}

Therefore, the area of the rectangle is:

• {eq}A = l\times w {/eq}
• {eq}A =20.23\times 14.7 {/eq}
• {eq}\color{blue}{A \approx 297.381\; \rm m^2} {/eq}

The perimeter is equal to:

• {eq}P = 2(l + w) {/eq}
• {eq}P = 2(20.23 + 14.7) {/eq}
• {eq}\color{blue}{P = 34.93\; \rm m} {/eq}