Let's break down and solve the given problem step by step:

Problem:

The diagram shows an isosceles triangle $PQR$ on a Cartesian plane with the coordinates:

• $P(-3, 2)$
• $Q(4, 1)$
• $R(1, 6)$

The tasks are:

1. (a) Determine the perimeter of triangle $PQR$.
2. (b) (i) Determine the coordinates of point $S$. (ii) Hence, calculate the length of $SQ$.

Solution:

Part (a): Perimeter of Triangle $PQR$

The perimeter of a triangle is the sum of the lengths of its sides. We can calculate the lengths of sides $PQ$, $PR$, and $QR$ using the distance formula:

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

• Length of $PQ$: $PQ = \sqrt{(4 - (-3))^2 + (1 - 2)^2} = \sqrt{(4 + 3)^2 + (-1)^2} = \sqrt{7^2 + (-1)^2} = \sqrt{49 + 1} = \sqrt{50} = 7.071$

• Length of $PR$: $PR = \sqrt{(1 - (-3))^2 + (6 - 2)^2} = \sqrt{(1 + 3)^2 + (6 - 2)^2} = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 5.657$

• Length of $QR$: $QR = \sqrt{(1 - 4)^2 + (6 - 1)^2} = \sqrt{(-3)^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} = 5.831$

Thus, the perimeter is: $\text{Perimeter of } PQR = PQ + PR + QR = 7.071 + 5.657 + 5.831 = 18.559 \ (\text{approximately})$

Part (b): Coordinates of Point $S$ and Length of $SQ$

(i) Coordinates of point $S$:
The point $S$ lies on the perpendicular bisector of line $PQ$. To find $S$, we first need the midpoint $M$ of $PQ$ and the slope of the perpendicular bisector.

• Midpoint $M$ of $PQ$: $M = \left( \frac{-3 + 4}{2}, \frac{2 + 1}{2} \right) = \left( \frac{1}{2}, \frac{3}{2} \right) = (0.5, 1.5)$

• Slope of line $PQ$: $\text{slope of } PQ = \frac{1 - 2}{4 - (-3)} = \frac{-1}{7} = -\frac{1}{7}$

• Slope of the perpendicular bisector:
  The perpendicular slope is the negative reciprocal of the slope of $PQ$: $\text{Slope of the perpendicular bisector} = 7$

• Using the point-slope form of the line equation $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is the midpoint: $y - 1.5 = 7(x - 0.5)$ Expanding this equation: $y - 1.5 = 7x - 3.5$ $y = 7x - 2$

Thus, the equation of the perpendicular bisector is $y = 7x - 2$.

The coordinates of point $S$ are given as $(-2, 4)$, which lies on this line: $4 = 7(-2) - 2$ This holds true, so the coordinates of $S$ are $(-2, 4)$.

(ii) Length of $SQ$:
Using the distance formula again, we calculate the distance between points $S(-2, 4)$ and $Q(4, 1)$: $SQ = \sqrt{(4 - (-2))^2 + (1 - 4)^2} = \sqrt{(4 + 2)^2 + (-3)^2} = \sqrt{6^2 + (-3)^2} = \sqrt{36 + 9} = \sqrt{45} = 6.708$

Final Answers:

• (a) The perimeter of triangle $PQR$ is approximately 18.56 units.
• (b)(i) The coordinates of point $S$ are $(-2, 4)$.
• (b)(ii) The length of $SQ$ is approximately 6.71 units.

Do you have any questions or want more details on any step? Here are 5 related questions you might find useful:

1. How do you find the equation of a perpendicular bisector?
2. What is the distance formula and how does it work?
3. How can you determine the area of triangle $PQR$ on the Cartesian plane?
4. What is the significance of an isosceles triangle in coordinate geometry?
5. How do you find the coordinates of a point dividing a line segment in a given ratio?

Tip: Always verify calculations when working with coordinates, especially using the distance formula, to avoid small errors!