Video Transcript
𝐴𝐵𝐶𝐷 is a rhombus with a side length of 24 centimeters and an interior angle of 79 degrees. Find the area of the rhombus, giving the answer to the nearest hundredth.
Let’s begin by sketching 𝐴𝐵𝐶𝐷. It is a rhombus, so all of its four sides are of equal length. And we’re told in the question that their length is 24 centimeters. We’re also given that one of the interior angles in the rhombus is 79 degrees. So it looks like this. We’re asked to find the area of this rhombus. Now, a rhombus is just a special type of parallelogram. So we could calculate its area using the formula base multiplied by perpendicular height. We know the base length of this rhombus, but we don’t know its perpendicular height. We could calculate this using trigonometry, but there is a more efficient method we can use.
We can recall that the diagonals of a rhombus divide the rhombus into two congruent triangles. And we can prove this using the side-side-side congruency condition. In the two triangles 𝐴𝐵𝐷 and 𝐶𝐵𝐷, the side 𝐵𝐷 is shared between the two triangles. 𝐴𝐵 is the same length as 𝐶𝐵, as they’re each sides of the original rhombus. And for the same reason, 𝐴𝐷 is the same length as 𝐶𝐷. So using the side-side-side congruency condition, we’ve proved that the triangles 𝐴𝐵𝐷 and 𝐶𝐵𝐷 are congruent. The areas are therefore the same, and they’re each half the area of the rhombus 𝐴𝐵𝐶𝐷.
Let’s consider then how we could find the area of one of these triangles. We recall the trigonometric formula for the area of a triangle. In a triangle 𝐴𝐵𝐶, where the uppercase letters 𝐴, 𝐵, and 𝐶 represent the measures of the triangle’s three angles and the lowercase letters 𝑎, 𝑏, and 𝑐 represent the lengths of their opposite sides, the trigonometric formula for the area of a triangle is a half 𝑎𝑏 sin 𝐶. We don’t need to be overly concerned about the letters 𝐴, 𝐵, and 𝐶. But we need to remember that the letters 𝑎 and 𝑏 in the formula represent the lengths of any two sides and the uppercase 𝐶 is the measure of their included angle. That’s the angle between these two sides.
Returning to the rhombus, in triangle 𝐵𝐶𝐷, we see that we’ve got the lengths of the sides 𝐵𝐶 and 𝐶𝐷, and we know the measure of their included angle. So applying the trigonometric formula for the area of a triangle, we have that the area of triangle 𝐵𝐶𝐷 is equal to a half multiplied by 24 multiplied by 24 multiplied by sin of 79 degrees. We already explained why the area of the rhombus 𝐴𝐵𝐶𝐷 is twice the area of each triangle. So we have that the area of 𝐴𝐵𝐶𝐷 is two multiplied by a half multiplied by 24 multiplied by 24 multiplied by sin of 79 degrees.
Now, of course, the factor of two and the factor of a half will cancel one another out to leave 24 multiplied by 24 multiplied by sin of 79 degrees. We can now evaluate this on a calculator, ensuring that our calculator is in degree mode. And it gives 565.4172 continuing. The question asks us to give our answer to the nearest hundredth. So we need to round to two decimal places, which is 565.42. So by recalling that the diagonals of a rhombus divide it into two congruent triangles and then applying the trigonometric formula for the area of a triangle, we found that the area of this rhombus to the nearest hundredth is 565.42 square centimeters.
