https://file.aiquickdraw.com/mathbot/file/1725775612512fb9ppq8u.jpg

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!

The diagram shows an isosceles triangle PQR drawn on a Cartesian plane. Tasks include: (a) determining the perimeter of triangle PQR, (b) (i) determining the coordinates of point S, and (ii) calculating the length of SQ.

Solve the problem of determining the perimeter of an isosceles triangle PQR on a Cartesian plane, find the coordinates of point S on the perpendicular bisector of PQ, and calculate the length of SQ. Suitable for students in Grade 9-11, this exercise covers key concepts of coordinate geometry and triangle properties.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation