A. In a parallelogram, opposite sides are equal. This means that we have two sides equal to 7 inches and two sides equal to 9 inches. Let's denote one angle between the sides equal to 7 and 9 inches as $\alpha$. The parallelogram can be divided into two congruent triangles by drawing any of the diagonals. Let's draw the longer diagonal of 14 inches. Now we have two congruent triangles with sides 7, 9 and 14 inches.
We can use the Law of Cosines to find the angle $\alpha$:
$14^2 = 7^2 + 9^2 - 2(7)(9) \cos{\alpha}$
Solving for $\cos{\alpha}$:
$\cos{\alpha} = \frac{7^2 + 9^2 - 14^2}{(2)(7)(9)} = \frac{49+81-196}{126} = \frac{-66}{126} = -\frac{11}{21}$
Now that we know $\cos{\alpha}$, we can use the Law of Cosines again to find the length of the other diagonal $d$:
$d^2 = 7^2 + 9^2 - 2(7)(9)(-\frac{11}{21}) = 49 + 81 + \frac{1386}{21} = 130 + \frac{1386}{21} = 130 + 66 = 196$
So, the length of the shorter diagonal is $\sqrt{196} = 14$. No, we do not need a calculator to do this.
B. We are asked to evaluate $7^2 + 9^2 + 7^2 + 9^2 - 14^2 - d^2$. We just found that $d=14$, so:
$7^2 + 9^2 + 7^2 + 9^2 - 14^2 - 14^2 = 49 + 81 + 49 + 81 - 196 - 196 = 260 - 392 = -132$
This expression represents the sum of the squares of all four sides minus the squares of the two diagonals. It seems we've seen the Law of Cosines in the previous part of the problem. But since we now know the angle, we'd like to calculate -2 times the product of the sides times the cosine of the angle between them. Using the angle that we calculated earlier:
$-2(7)(9)\cos{\alpha} = -2(7)(9)(-\frac{11}{21}) = -\frac{1386}{21} = 66$. It is only one term, while the expression we had to evaluate was -132, which is 2 times smaller.
A.
A parallelogram has a 7-inch side and a 9-inch side, and the longer diagonal is 14 inches long. Find the length of the other diagonal. Do you need your calculator to do it?
B. (Continuation) Evaluate 72 + 92 + 72 + 92 − 142 − d2, where d is the length of the
shorter diagonal. Hmm. . .where have you seen this before?
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