To find the perimeter of the triangle with vertices [tex]\(P(6, 4)\)[/tex], [tex]\(Q(-3, 1)\)[/tex], and [tex]\(R(9, -5)\)[/tex], we need to calculate the distances between each pair of these points to determine the lengths of the sides of the triangle. Finally, we'll add up these lengths to find the perimeter.
### Step 1: Calculate the Distance Between Points [tex]\(P\)[/tex] and [tex]\(Q\)[/tex]
The coordinates of [tex]\(P\)[/tex] are [tex]\((6, 4)\)[/tex] and the coordinates of [tex]\(Q\)[/tex] are [tex]\((-3, 1)\)[/tex]. The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Plugging in the coordinates of [tex]\(P\)[/tex] and [tex]\(Q\)[/tex]:
[tex]\[
PQ = \sqrt{((-3) - 6)^2 + (1 - 4)^2} = \sqrt{(-9)^2 + (-3)^2} = \sqrt{81 + 9} = \sqrt{90}
\][/tex]
[tex]\[
PQ = \sqrt{9 \times 10} = 3\sqrt{10}
\][/tex]
### Step 2: Calculate the Distance Between Points [tex]\(Q\)[/tex] and [tex]\(R\)[/tex]
The coordinates of [tex]\(Q\)[/tex] are [tex]\((-3, 1)\)[/tex] and the coordinates of [tex]\(R\)[/tex] are [tex]\((9, -5)\)[/tex]. Using the distance formula again:
[tex]\[
QR = \sqrt{(9 - (-3))^2 + (-5 - 1)^2} = \sqrt{(9 + 3)^2 + (-5 - 1)^2} = \sqrt{12^2 + (-6)^2} = \sqrt{144 + 36} = \sqrt{180}
\][/tex]
[tex]\[
QR = \sqrt{36 \times 5} = 6\sqrt{5}
\][/tex]
### Step 3: Calculate the Distance Between Points [tex]\(R\)[/tex] and [tex]\(P\)[/tex]
The coordinates of [tex]\(R\)[/tex] are [tex]\((9, -5)\)[/tex] and the coordinates of [tex]\(P\)[/tex] are [tex]\((6, 4)\)[/tex]. Again using the distance formula:
[tex]\[
RP = \sqrt{(6 - 9)^2 + (4 - (-5))^2} = \sqrt{(6 - 9)^2 + (4 + 5)^2} = \sqrt{(-3)^2 + 9^2} = \sqrt{9 + 81} = \sqrt{90}
\][/tex]
[tex]\[
RP = \sqrt{9 \times 10} = 3\sqrt{10}
\][/tex]
### Step 4: Calculate the Perimeter of the Triangle
Now we add up the lengths of the sides [tex]\(PQ\)[/tex], [tex]\(QR\)[/tex], and [tex]\(RP\)[/tex] to find the perimeter:
[tex]\[
\text{Perimeter} = PQ + QR + RP = 3\sqrt{10} + 6\sqrt{5} + 3\sqrt{10}
\][/tex]
[tex]\[
\text{Perimeter} = 3\sqrt{10} + 3\sqrt{10} + 6\sqrt{5} = 6\sqrt{10} + 6\sqrt{5}
\][/tex]
So, the perimeter of the triangle in surd form is:
[tex]\[
\boxed{6\sqrt{10} + 6\sqrt{5}}
\][/tex]
