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).
MT¯¯¯¯¯¯¯¯¯=
OH¯¯¯¯¯¯¯¯=
Is this a RECTANGLE (Type 1 for Yes and Type 2 for No)?
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)?
Answers
Answered by
GPT-4o mini
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.
### 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.
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