Question

The coordinates of the vertices of △PQR are given. P(−4, 1) Q(2, 9) R(8, 1) Give the length of each side and what is the perimeter of △PQR ? (4 points)

Answer 1

To find the length of each side of triangle \( \triangle PQR \) with vertices \( P(-4, 1) \), \( Q(2, 9) \), and \( R(8, 1) \), we can use the distance formula, which is given by:

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

### Length of Side PQ

Using points \( P(-4, 1) \) and \( Q(2, 9) \):

\[
PQ = \sqrt{(2 - (-4))^2 + (9 - 1)^2}
\]
\[
= \sqrt{(2 + 4)^2 + (9 - 1)^2}
\]
\[
= \sqrt{(6)^2 + (8)^2}
\]
\[
= \sqrt{36 + 64}
\]
\[
= \sqrt{100}
\]
\[
= 10
\]

### Length of Side QR

Using points \( Q(2, 9) \) and \( R(8, 1) \):

\[
QR = \sqrt{(8 - 2)^2 + (1 - 9)^2}
\]
\[
= \sqrt{(6)^2 + (-8)^2}
\]
\[
= \sqrt{36 + 64}
\]
\[
= \sqrt{100}
\]
\[
= 10
\]

### Length of Side PR

Using points \( P(-4, 1) \) and \( R(8, 1) \):

\[
PR = \sqrt{(8 - (-4))^2 + (1 - 1)^2}
\]
\[
= \sqrt{(8 + 4)^2 + (0)^2}
\]
\[
= \sqrt{(12)^2}
\]
\[
= 12
\]

### Perimeter of Triangle PQR

Now, we can find the perimeter \( P \) of triangle \( PQR \) by adding the lengths of its sides:

\[
P = PQ + QR + PR
\]
\[
= 10 + 10 + 12
\]
\[
= 32
\]

### Final Answers

- Length of side \( PQ = 10 \)
- Length of side \( QR = 10 \)
- Length of side \( PR = 12 \)
- Perimeter of \( \triangle PQR = 32 \)