Question

The following veritces form a parallelogram. Do the following vertices form a RECTANGLE also? Prove using the theorem that says IF the diagonals are congruent, then it is a rectanlge. M(−7,6), O(−2,9), T(4,−1), H(−1,−4) (2 points) Round to the tenths place as needed (2 decimal places).

Answer 1

To determine if the given vertices form a rectangle, we first need to confirm that the diagram is a parallelogram and then check if the diagonals are congruent. The vertices provided are \( M(-7,6) \), \( O(-2,9) \), \( T(4,-1) \), and \( H(-1,-4) \).

### Step 1: Analyze the Position of Points

First, let's determine the midpoints of the diagonals \( MT \) and \( OH \) to see if they are the same, which would confirm that they are indeed the diagonals of the parallelogram.

#### Diagonal \( MT \)
1. Coordinates of \( M \): \( (-7, 6) \)
2. Coordinates of \( T \): \( (4, -1) \)

The midpoint \( M_{MT} \) of \( MT \) is calculated as follows:

\[
M_{MT} = \left( \frac{-7 + 4}{2}, \frac{6 + (-1)}{2} \right) = \left( \frac{-3}{2}, \frac{5}{2} \right) = (-1.5, 2.5)
\]

#### Diagonal \( OH \)
1. Coordinates of \( O \): \( (-2, 9) \)
2. Coordinates of \( H \): \( (-1, -4) \)

The midpoint \( M_{OH} \) of \( OH \) is calculated as follows:

\[
M_{OH} = \left( \frac{-2 + (-1)}{2}, \frac{9 + (-4)}{2} \right) = \left( \frac{-3}{2}, \frac{5}{2} \right) = (-1.5, 2.5)
\]

Since \( M_{MT} = M_{OH} \), \( MT \) and \( OH \) are indeed the diagonals of a parallelogram.

### Step 2: Calculate the Lengths of the Diagonals

Next, we will calculate the lengths of the diagonals \( MT \) and \( OH \) to see if they are congruent.

#### Length of Diagonal \( MT \)

Using the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

For \( M(-7, 6) \) and \( T(4, -1) \):

\[
MT = \sqrt{(4 - (-7))^2 + (-1 - 6)^2} = \sqrt{(4 + 7)^2 + (-1 - 6)^2} = \sqrt{11^2 + (-7)^2} = \sqrt{121 + 49} = \sqrt{170} \approx 13.0
\]

#### Length of Diagonal \( OH \)

For \( O(-2, 9) \) and \( H(-1, -4) \):

\[
OH = \sqrt{(-1 - (-2))^2 + (-4 - 9)^2} = \sqrt{(1)^2 + (-13)^2} = \sqrt{1 + 169} = \sqrt{170} \approx 13.0
\]

### Conclusion

Since both diagonals \( MT \) and \( OH \) are congruent (both have a length of approximately 13.0), we can conclude that the given vertices form a rectangle.

**Final Answer:** Yes, the vertices form a rectangle.