Question

In a right triangle, the hypotenuse has endpoints P(–3, 2) and Q(1, –3).

On a coordinate plane, line P Q has points (negative 3, 2) and (1, negative 3).

If R represents the third vertex in the triangle and R is located in the third quadrant, what is the length of PR?
3
4
5
6

Answer 1

To find the length of \( PR \), we first need to determine the coordinates of points \( P \) and \( Q \) and then validate the properties of the triangle, including the position of the vertex \( R \).

The coordinates of points \( P \) and \( Q \) are:
- \( P(-3, 2) \)
- \( Q(1, -3) \)

Next, we can calculate the length of the hypotenuse \( PQ \) using the distance formula:

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

Substituting the coordinates of \( P \) and \( Q \):

\[
PQ = \sqrt{(1 - (-3))^2 + (-3 - 2)^2}
\]
\[
= \sqrt{(1 + 3)^2 + (-3 - 2)^2}
\]
\[
= \sqrt{4^2 + (-5)^2}
\]
\[
= \sqrt{16 + 25}
\]
\[
= \sqrt{41}
\]

Now, since we are forming a right triangle with the hypotenuse \( PQ \) and the vertex \( R \) is located in the third quadrant, which has negative \( x \) and \( y \) coordinates, we can set \( R(x_R, y_R) \).

In a right triangle, the two legs must be perpendicular. Given points \( P \) and \( Q \), point \( R \) can lie anywhere in the third quadrant fulfilling this condition. The general condition of a right triangle states that the lengths of the legs can be found using the Pythagorean theorem which can be represented as follows with \( PR \) as one of the legs and \( QR \) as another:

\[
PR^2 + QR^2 = PQ^2
\]

However, without specific coordinates for \( R \), we cannot compute the exact lengths.

But from the choices provided (3, 4, 5, or 6), we notice that:
- A possible integer length for \( PR \) can be checked against commonly known right triangles.

To assess probable distances:
- Checking for \( PR = 5 \):
\[
QR^2 = PQ^2 - PR^2 = 41 - 25 = 16
\]
So \( QR = 4 \) (possible integers, fulfilling the right triangle condition).

By picking \( PR = 5 \), both coordinates \( PR\) and \( QR \) still allow for \( R \) to indeed lie in the third quadrant.

Thus, the length of \( PR \) is \( \boxed{5} \).