The area of a parallelogram is $12$ , and the lengths of its sides are $7$ . $4$ and $4$ . $7$ . Determine, to the nearest tenth of a degree, the measure of the acute angle of the parallelogram. $\newline$ Answer:

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Grade 8

Find missing angles in quadrilaterals I

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Q. The area of a parallelogram is

12

, and the lengths of its sides are

7

4

and

4

7

. Determine, to the nearest tenth of a degree, the measure of the acute angle of the parallelogram.

\newline

Answer:

Choose Base Side: The area of a parallelogram is given by the formula: Area = base $\times$ height. In this case, we can consider one of the sides as the base. Let's choose the side with length $7.4$ as the base.
Find Height: Now, we need to find the height of the parallelogram. The height is the perpendicular distance from the base to the opposite side. We can use the area formula to solve for the height $h$ : $\text{Area} = \text{base} \times \text{height}$ , which gives us $h = \frac{\text{Area}}{\text{base}}$ .
Calculate Height: Plugging in the values we have: $h = \frac{12}{7.4}$ . Let's calculate the height. $\newline$ $h = \frac{12}{7.4} \approx 1.6216$
Use Trigonometry: The height we've found is the side length of the parallelogram when projected perpendicularly onto the base. This height forms a right triangle with the side of the parallelogram ( $4.7$ ) and the acute angle we are trying to find.
Calculate Cosine: To find the measure of the acute angle, we can use the trigonometric function cosine, which is defined as $\frac{\text{adjacent}}{\text{hypotenuse}}$ in a right triangle. In this case, the adjacent side is the height ( $h$ ), and the hypotenuse is the side of the parallelogram ( $4.7$ ).
Find Acute Angle: The cosine of the acute angle $\theta$ is therefore $\cos(\theta) = \frac{h}{4.7}$ . Plugging in the value of $h$ we found: $\cos(\theta) = \frac{1.6216}{4.7}$ .
Calculate Angle: Calculating the cosine of the angle gives us: $\cos(\theta) \approx \frac{1.6216}{4.7} \approx 0.3450$ .
Calculate Angle: Calculating the cosine of the angle gives us: $\cos(\theta) \approx \frac{1.6216}{4.7} \approx 0.3450$ .To find the angle $\theta$ , we need to take the inverse cosine (arccos) of $0.3450$ . Let's calculate $\theta = \arccos(0.3450)$ .
Calculate Angle: Calculating the cosine of the angle gives us: $\cos(\theta) \approx \frac{1.6216}{4.7} \approx 0.3450$ .To find the angle $\theta$ , we need to take the inverse cosine (arccos) of $0.3450$ . Let's calculate $\theta = \arccos(0.3450)$ .After calculating, we find that $\theta \approx \arccos(0.3450) \approx 69.8^\circ$ to the nearest tenth of a degree.