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### Problem 4
#### a. Find the gradient of the line PQ.
Given points \( P(-2, 6) \) and \( Q(6, 2) \).
The gradient \( m \) is calculated by:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
\[
m = \frac{2 - 6}{6 - (-2)} = \frac{-4}{8} = -\frac{1}{2}
\]

#### b. If angle RPQ is a right angle, what is the gradient of the line PR?
Since \( PQ \) and \( PR \) are perpendicular, the product of their gradients is \(-1\). Given \( m_{PQ} = -\frac{1}{2} \),
\[
m_{PR} \cdot m_{PQ} = -1 \Rightarrow m_{PR} \cdot \left(-\frac{1}{2}\right) = -1 \Rightarrow m_{PR} = 2
\]

#### c. Use your answer from (b), or otherwise, to find the value of \( a \).
For \( PR \), given \( R(8, a) \) and \( P(-2, 6) \), using the gradient formula:
\[
m_{PR} = \frac{a - 6}{8 - (-2)} = \frac{a - 6}{10}
\]
Setting \( m_{PR} = 2 \),
\[
\frac{a - 6}{10} = 2 \Rightarrow a - 6 = 20 \Rightarrow a = 26
\]

#### d. Calculate the distance PQ.
Using the distance formula,
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
\[
d = \sqrt{(6 - (-2))^2 + (2 - 6)^2} = \sqrt{8^2 + (-4)^2} = \sqrt{64 + 16} = \sqrt{80} = 4\sqrt{5}
\]

#### e. Calculate the midpoint of QR.
Given \( Q(6, 2) \) and \( R(8, a) \), with \( a = 26 \),
\[
M = \left(\frac{6 + 8}{2}, \frac{2 + 26}{2}\right) = (7, 14)
\]

### Problem 5
#### a. Calculate the values of \( s \) and \( t \).
Given \( A(s, 8) \) to \( B(-2, t) \) with midpoint \( M(2, 3) \).
Using the midpoint formula,
\[
\frac{s + (-2)}{2} = 2 \Rightarrow s - 2 = 4 \Rightarrow s = 6
\]
\[
\frac{8 + t}{2} = 3 \Rightarrow 8 + t = 6 \Rightarrow t = -2
\]

#### b. Find the equation of the straight line perpendicular to AB, passing through the point M.
Gradient of \( AB \):
\[
m_{AB} = \frac{t - 8}{-2 - s} = \frac{-2 - 8}{-2 - 6} = \frac{-10}{-8} = \frac{5}{4}
\]
The perpendicular gradient is \(-\frac{4}{5}\). Using point-slope form,
\[
y - 3 = -\frac{4}{5}(x - 2) \Rightarrow y = -\frac{4}{5}x + \frac{8}{5} + 3 = -\frac{4}{5}x + \frac{23}{5}
\]
\[
y = -\frac{4}{5}x + \frac{23}{5}
\]

Would you like more details on any step, or do you have any other questions? Here are some related questions to consider:

1. How do you verify the right angle in a triangle using slopes?
2. What is the equation of the circle passing through points P, Q, and R?
3. How do you determine if three points are collinear?
4. Can you find the area of triangle PQR?
5. How does the distance formula derive from the Pythagorean theorem?

**Tip:** Remember to verify your calculations by plugging values back into the original equations to ensure consistency and correctness.

The coordinates of the vertices of a triangle are P(-2, 6), Q(6, 2), and R(-8, a). Find the gradient of line PQ, determine the gradient of PR if angle RPQ is a right angle, use this to find the value of a, calculate the distance PQ, and find the midpoint of QR. Also, for a line joining A(s, 8) to B(-2, t) with a midpoint M(2, 3), calculate the values of s and t, and find the equation of the line perpendicular to AB passing through M.

This problem involves calculating gradients, distances, midpoints, and solving for unknowns in a triangle with vertices P(-2, 6), Q(6, 2), and R(-8, a). It also covers perpendicular line equations and uses core geometry concepts like the gradient and distance formulas. Suitable for high school students in Grade 10-12.

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