Question

If the points $P(7,5)$, $Q(a,2a)$ and $R(12,30)$ are collinear, then the value of a is equal to

• A. 5
• B. 6
• C. 8
• D. 9
• E. 10

Hint:

Geometry Tip: You can also use the area of a triangle formula for collinear points! Set the area determinant $\frac{1}{2}|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)| = 0$ to solve for unknowns.

Answer 1

Answer: (E)

Concept:
For three points to be collinear, they must lie on the exact same straight line. This means the slope of the line segment connecting any two of the points must be perfectly equal to the slope of the segment connecting another two. The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$.

Step 1: Calculate the slope of segment PR.
Using points $P(7, 5)$ and $R(12, 30)$: $$m_{PR} = \frac{30 - 5}{12 - 7}$$ $$m_{PR} = \frac{25}{5} = 5$$

Step 2: Express the slope of segment PQ.
Using points $P(7, 5)$ and $Q(a, 2a)$: $$m_{PQ} = \frac{2a - 5}{a - 7}$$

Step 3: Set the slopes equal to each other.
Because the points are collinear, the slope of PQ must equal the slope of PR ($m_{PQ} = m_{PR}$): $$\frac{2a - 5}{a - 7} = 5$$

Step 4: Solve the algebraic equation.
Multiply both sides by $(a - 7)$ to eliminate the fraction: $$2a - 5 = 5(a - 7)$$ Distribute the 5 on the right side: $$2a - 5 = 5a - 35$$

Step 5: Isolate the variable a.
Subtract $2a$ from both sides and add $35$ to both sides: $$35 - 5 = 5a - 2a$$ $$30 = 3a$$ $$a = 10$$ Hence the correct answer is (E) 10.