Coordinate Geometry Questions and Answers

Answer: A

Let's find the slopes of the lines connecting the points.

Slope between (2,-2) and (-2,1) = \(\frac{1-(-2)}{-2-2} = \frac{3}{-4}\).

Slope between (-2,1) and (5,2) = \(\frac{2-1}{5-(-2)} = \frac{1}{7}\).

Slope between (5,2) and (2,-2) = \(\frac{-2-2}{2-5} = \frac{-4}{-3} = \frac{4}{3}\).

The product of the slopes of the first and third lines is \((-\frac{3}{4}) \times (\frac{4}{3}) = -1\). Since the product is -1, these two lines are perpendicular, forming a right angle. Therefore, the triangle is a right-angled triangle.

Answer: A

Use the point-slope form.

\[ y - 6 = -\frac{4}{3}(x + 3) \]

\[ 3y - 18 = -4x - 12 \quad\Longrightarrow\quad 4x + 3y = 6 \]

Checking: at \( (-3, 6) \), \( -12 + 18 = 6 \).

Answer: B

The line 4x - y = 1 has slope 4, so a perpendicular line has slope -1/4. Through (2, -5) its equation is

The option 4x - y + 18 = 0 is parallel to the given line (same slope 4), x - 4y - 18 = 0 has slope 1/4, which is neither, and 4x + 3y - 9 = 0 has slope -4/3. Only x + 4y + 18 = 0 also passes through (2, -5): 2 - 20 + 18 = 0.

Answer: C

The value of (p - q) is 3.

The slope through (4, p) and (0, q) is the change in y over the change in x:

\[ \begin{aligned} \frac{p - q}{4 - 0} &= \frac{3}{4} \\ p - q &= 3 \end{aligned} \]

-3 comes from taking the points in the opposite order for y but not for x. 0 would mean a horizontal line, and 4 confuses the run with the rise.