Question

The top right corner p
The top left corner q
The bottom right corner t
The bottom left corner r
There is a line connecting to q and t

The length of the rectangle is 15
The side is eight

Which statement proves that parallelogram PORT is a rectangle?

15 ^ 2 - 8 ^ 2 = T * rho ^ 2

8 + 15 = TQ

15 - 8 = TQ

8 ^ 2 + 15 ^ 2 = T * rho ^ 2

Answer 1

To determine which statement proves that parallelogram PORT is a rectangle, we need to recall the properties of rectangles and parallelism in parallelograms.

In a rectangle, the diagonals are equal in length. Therefore, to show that parallelogram PORT is a rectangle, we can use the relationship involving the lengths of the sides and the diagonals.

Given that the length of rectangle (side opposite to the width) = 15 and the width = 8, we can compute the length of the diagonal using the Pythagorean theorem. The diagonal (let’s denote it as d) can be calculated as:

\[
d = \sqrt{(15^2 + 8^2)}
\]

Now, if we look at the potential statements provided:

1. **15^2 - 8^2 = T * rho^2**
2. **8 + 15 = TQ**
3. **15 - 8 = TQ**
4. **8^2 + 15^2 = T * rho^2**

We focus specifically on statement 4:

**8^2 + 15^2 = T * rho^2.**

This expression resembles the Pythagorean theorem’s relation (since we hypothesized that the diagonal d follows the Pythagorean theorem).

**Calculating based on rectangle properties:**

- The equation \(8^2 + 15^2\) gives us the length of the diagonal squared, confirming if diagonals are equal (which must be validated by \(2T^2\) for arguments sake).
- This statement can be a reasonable and relevant proof since it uses the properties of lengths associated with diagonals in determining if it is a rectangle by its dimensions.

Therefore, the statement that proves parallelogram PORT is a rectangle is:

**8^2 + 15^2 = T * rho^2**

This is arguably the closest to demonstrating the right angles and confirming equal diagonal lengths inherent to rectangles.