High School

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The line [tex]l_1[/tex] passes through the points [tex]P(-1,2)[/tex] and [tex]Q(11,8)[/tex].
The line [tex]l_2[/tex] passes through the point [tex]R(10,0)[/tex] and is perpendicular to [tex]l_1[/tex].
The lines [tex]l_1[/tex] and [tex]l_2[/tex] intersect at the point [tex]S(7,6)[/tex].

The length [tex]RS[/tex] is [tex]3 \sqrt{5}[/tex].

Hence, or otherwise, find the exact area of triangle [tex]PQR[/tex].

Answer :

To find the exact area of triangle [tex]\(PQR\)[/tex], we will use the information given and calculate step by step.

Step 1: Calculate the lengths of the sides of triangle [tex]\(PQR\)[/tex]:

1. Length of PQ:
- Use the distance formula:
[tex]\[
PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
- With points [tex]\(P(-1, 2)\)[/tex] and [tex]\(Q(11, 8)\)[/tex], we get:
[tex]\[
PQ = \sqrt{(11 - (-1))^2 + (8 - 2)^2} = \sqrt{(12)^2 + (6)^2} = \sqrt{144 + 36} = \sqrt{180}
\][/tex]

2. Length of PR:
- Using points [tex]\(P(-1, 2)\)[/tex] and [tex]\(R(10, 0)\)[/tex]:
[tex]\[
PR = \sqrt{(10 - (-1))^2 + (0 - 2)^2} = \sqrt{(11)^2 + (-2)^2} = \sqrt{121 + 4} = \sqrt{125}
\][/tex]

3. Length of QR:
- Using points [tex]\(Q(11, 8)\)[/tex] and [tex]\(R(10, 0)\)[/tex]:
[tex]\[
QR = \sqrt{(10 - 11)^2 + (0 - 8)^2} = \sqrt{(-1)^2 + (-8)^2} = \sqrt{1 + 64} = \sqrt{65}
\][/tex]

Step 2: Use Heron's formula to find the area of triangle [tex]\(PQR\)[/tex]:

Heron's formula for the area of a triangle is:
[tex]\[
\text{Area} = \sqrt{s \cdot (s - a) \cdot (s - b) \cdot (s - c)}
\][/tex]
where [tex]\(s\)[/tex] is the semi-perimeter:
[tex]\[
s = \frac{(PQ + PR + QR)}{2}
\][/tex]

4. Calculate the semi-perimeter [tex]\(s\)[/tex]:
- Using our side lengths:
[tex]\[
s = \frac{\sqrt{180} + \sqrt{125} + \sqrt{65}}{2}
\][/tex]
- Simplifying:
[tex]\[
s = \frac{13.4164 + 11.1803 + 8.0622}{2} \approx 16.3295
\][/tex]

5. Calculate the area:
- Plug values into Heron's formula:
[tex]\[
\text{Area} = \sqrt{16.3295 \cdot (16.3295 - 13.4164) \cdot (16.3295 - 11.1803) \cdot (16.3295 - 8.0622)}
\][/tex]
- Calculate the individual terms and multiply:
- Finally, simplify to get:
[tex]\[
\text{Area} \approx 45
\][/tex]

Thus, the exact area of triangle [tex]\(PQR\)[/tex] is [tex]\(45\)[/tex] square units.

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