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.

Question

M(−7,6), O(−2,9), T(4,−1), H(−1,−4)


(2 points)
Round to the tenths place as needed (2 decimal places).

MT¯¯¯¯¯¯¯¯¯=

OH¯¯¯¯¯¯¯¯=



Is this a RECTANGLE (Type 1 for Yes and Type 2 for No)?

Answer 1

To determine if the given vertices \( M(-7,6) \), \( O(-2,9) \), \( T(4,-1) \), and \( H(-1,-4) \) form a rectangle, we will first find the lengths of the diagonals \( MT \) and \( OH \), and then check if they are congruent.

Step 1: Calculate the lengths of the diagonals

Finding the length of diagonal \( MT \):

The length of the segment between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) can be calculated using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

For points \( M(-7, 6) \) and \( T(4, -1) \): \[ MT = \sqrt{(4 - (-7))^2 + (-1 - 6)^2} \] \[ = \sqrt{(4 + 7)^2 + (-1 - 6)^2} \] \[ = \sqrt{(11)^2 + (-7)^2} \] \[ = \sqrt{121 + 49} \] \[ = \sqrt{170} \] \[ = 13.0384 \quad (\text{rounded to the tenths place is } 13.0) \]

Finding the length of diagonal \( OH \)

For points \( O(-2, 9) \) and \( H(-1, -4) \): \[ OH = \sqrt{(-1 - (-2))^2 + (-4 - 9)^2} \] \[ = \sqrt{(-1 + 2)^2 + (-4 - 9)^2} \] \[ = \sqrt{(1)^2 + (-13)^2} \] \[ = \sqrt{1 + 169} \] \[ = \sqrt{170} \] \[ = 13.0384 \quad (\text{rounded to the tenths place is } 13.0) \]

Summary of results

• The length of diagonal \( MT \) is \( 13.0 \)
• The length of diagonal \( OH \) is \( 13.0 \)

Step 2: Check if the diagonals are congruent

Since both diagonals \( MT \) and \( OH \) have the same length (both are \( 13.0 \)), they are congruent.

Conclusion

According to the theorem, if the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

Thus, the answer is:

• MT = 13.0
• OH = 13.0
• Is this a RECTANGLE? Type 1 for Yes.