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?

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. To find the length of the other diagonal of the parallelogram, we can use the properties of a parallelogram.

In a parallelogram, opposite sides are equal in length and opposite angles are equal.

Given that the parallelogram has a 7-inch side and a 9-inch side, we know that the two opposite sides of the parallelogram are 7 inches and 9 inches.

We are also given that the longer diagonal is 14 inches long.

Using the properties of a parallelogram, we can determine that the shorter diagonal is also 14 inches long. This is because the diagonals of a parallelogram bisect each other, meaning they divide each other into two equal segments.

Therefore, the length of the other diagonal of the parallelogram is also 14 inches.

No, you do not need a calculator to find the length of the other diagonal. By understanding the properties of a parallelogram, you can find the answer without performing any calculations.

B. The expression 72 + 92 + 72 + 92 - 142 - d2 can be simplified further.

The square of the length of the shorter diagonal (d) is d2.

Knowing that the length of the shorter diagonal is 14 inches (as explained in the previous question), we can substitute d with 14 in the expression to get:

72 + 92 + 72 + 92 - 142 - 142

Calculating the expression, we have:

49 + 81 + 49 + 81 - 196 - 196

Simplifying further:

(49 + 49) + (81 + 81) - (196 + 196)

98 + 162 - 392

260 - 392

-132

Therefore, the simplified value of 72 + 92 + 72 + 92 - 142 - d2, where d is the length of the shorter diagonal, is -132.

As for where you might have seen this expression before, it resembles the formula for the Pythagorean Theorem, which states that a² + b² = c², where a and b are the lengths of the two sides of a right triangle, and c is the length of the hypotenuse.