Prove that triangle with vertices P(2,-4),Q(-4,4) and R(3,3) is an isosceles triangle.Draw triangle by plotting the points on the Cartesian plan and Prove by calculation?

To prove that triangle PQR is an isosceles triangle, we need to show that at least two sides of the triangle are equal in length.

First, let's plot the points P(2,-4), Q(-4,4), and R(3,3) on the Cartesian plane:

Now, let's calculate the lengths of the sides of the triangle:

1. Length of side PQ:
Using the distance formula, we have:
PQ = sqrt((x2 - x1)^2 + (y2 - y1)^2)
PQ = sqrt((-4 - 2)^2 + (4 - (-4))^2)
PQ = sqrt((-6)^2 + (8)^2)
PQ = sqrt(36 + 64)
PQ = sqrt(100)
PQ = 10

2. Length of side QR:
Using the distance formula, we have:
QR = sqrt((x2 - x1)^2 + (y2 - y1)^2)
QR = sqrt((3 - (-4))^2 + (3 - 4)^2)
QR = sqrt((7)^2 + (-1)^2)
QR = sqrt(49 + 1)
QR = sqrt(50)
QR = 5√2

3. Length of side PR:
Using the distance formula, we have:
PR = sqrt((x2 - x1)^2 + (y2 - y1)^2)
PR = sqrt((3 - 2)^2 + (3 - (-4))^2)
PR = sqrt((1)^2 + (7)^2)
PR = sqrt(1 + 49)
PR = sqrt(50)
PR = 5√2

Since PQ = PR, we can conclude that triangle PQR is an isosceles triangle.

Plotting the Points and Drawing the Triangle

Plot the points P(2,-4), Q(-4,4), and R(3,3) on the Cartesian plane:

[Image of the triangle PQR on the Cartesian plane]

Calculating the Distances

Calculate the distances between each pair of points using the distance formula:

```
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
```

- Distance between P and Q:
```
PQ = sqrt((-4 - 2)^2 + (4 - (-4))^2) = sqrt(36 + 64) = sqrt(100) = 10
```
- Distance between P and R:
```
PR = sqrt((3 - 2)^2 + (3 - (-4))^2) = sqrt(1 + 49) = sqrt(50) = 5√2
```
- Distance between Q and R:
```
QR = sqrt((3 - (-4))^2 + (3 - 4)^2) = sqrt(49 + 1) = sqrt(50) = 5√2
```

Proof of Isosceles Triangle

An isosceles triangle has two equal sides. In this case, we have:

```
PR = QR = 5√2
```

Therefore, triangle PQR is an isosceles triangle with PR and QR as equal sides.