To determine the type of triangle \( \Delta ABC \) given the midpoints \( P(3, 3) \), \( Q(3, 4) \), and \( R(2, 4) \), we will follow these steps:
### Step 1: Identify the midpoints and their coordinates
The midpoints of the sides of triangle \( ABC \) are given as:
- \( P(3, 3) \)
- \( Q(3, 4) \)
- \( R(2, 4) \)
### Step 2: Calculate the slopes of the sides of triangle \( PQR \)
To find the slopes of the sides \( PQ \), \( PR \), and \( QR \), we will use the slope formula:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]
#### Slope of \( PQ \)
Using points \( P(3, 3) \) and \( Q(3, 4) \):
\[
\text{slope of } PQ = \frac{4 - 3}{3 - 3} = \frac{1}{0} \rightarrow \text{undefined (vertical line)}
\]
This means \( PQ \) is parallel to the y-axis.
#### Slope of \( PR \)
Using points \( P(3, 3) \) and \( R(2, 4) \):
\[
\text{slope of } PR = \frac{4 - 3}{2 - 3} = \frac{1}{-1} = -1
\]
#### Slope of \( QR \)
Using points \( Q(3, 4) \) and \( R(2, 4) \):
\[
\text{slope of } QR = \frac{4 - 4}{2 - 3} = \frac{0}{-1} = 0
\]
This means \( QR \) is parallel to the x-axis.
### Step 3: Analyze the triangle \( PQR \)
From the slopes calculated:
- \( PQ \) is vertical.
- \( QR \) is horizontal.
Since one side is vertical and another is horizontal, triangle \( PQR \) is a right triangle.
### Step 4: Determine the lengths of the sides of triangle \( PQR \)
Now, we will calculate the lengths of the sides to check if triangle \( PQR \) is isosceles.
#### Length of \( PQ \)
\[
PQ = \sqrt{(3 - 3)^2 + (4 - 3)^2} = \sqrt{0 + 1} = 1
\]
#### Length of \( QR \)
\[
QR = \sqrt{(3 - 2)^2 + (4 - 4)^2} = \sqrt{1 + 0} = 1
\]
#### Length of \( PR \)
\[
PR = \sqrt{(3 - 2)^2 + (4 - 3)^2} = \sqrt{1 + 1} = \sqrt{2}
\]
### Step 5: Conclusion about triangle \( ABC \)
Since \( PQ = QR = 1 \) and \( PR = \sqrt{2} \), two sides of triangle \( PQR \) are equal, confirming that it is an isosceles right triangle.
Since \( PQR \) is a right triangle and isosceles, triangle \( ABC \) must also be a right triangle and isosceles.
### Final Answer:
Triangle \( ABC \) is a right-angled isosceles triangle.
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