The (x,y) coordinates of vertices P, Q and R of a parallelogram PQRS are (−3,−2),(1,−5) and (9,1),respectively.IfthediagonalSQintersectsthex - axis at (a,0),thenthevalueofa$ is
⌨️ Shortcuts:[1-4 / A-D] Select[Enter] Check[B] Star[S] Solution
Official Correct Answer: C. ### Core Concept
In a parallelogram, the diagonals bisect each other. This means the midpoint of diagonal PR is the same as the midpoint of diagonal SQ.
### Step-by-Step Solution
1. **Find Midpoint of PR**:
- P(-3, -2) and R(9, 1)
- Midpoint formula: $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$
- Midpoint of PR: $\left( \frac{-3 + 9}{2}, \frac{-2 + 1}{2} \right) = (3, -0.5)$
2. **Determine Coordinates of S**:
- Midpoint of SQ must also be (3, -0.5)
- Let S = (s_x, s_y)
- Midpoint of SQ: $\left( \frac{1 + s_x}{2}, \frac{-5 + s_y}{2} \right) = (3, -0.5)$
- Solve for s_x and s_y:
- $\frac{1 + s_x}{2} = 3 \Rightarrow s_x = 5$
- $\frac{-5 + s_y}{2} = -0.5 \Rightarrow s_y = 4$
- S = (5, 4)
3. **Find Equation of Line SQ**:
- Points S(5, 4) and Q(1, -5)
- Slope (m): $\frac{-5 - 4}{1 - 5} = \frac{-9}{-4} = \frac{9}{4}$
- Using point-slope form with Q(1, -5):
- $y - (-5) = \frac{9}{4}(x - 1)$
- $y + 5 = \frac{9}{4}(x - 1)$
4. **Find Intersection with x-axis (y = 0)**:
- Substitute y = 0:
- $0 + 5 = \frac{9}{4}(x - 1)$
- $5 = \frac{9}{4}(x - 1)$
- Multiply both sides by 4: $20 = 9(x - 1)$
- $20 = 9x - 9 \Rightarrow 29 = 9x \Rightarrow x = \frac{29}{9}$
### 30-Second Shortcut
Since diagonals bisect each other, find S using midpoint. Then, use two-point formula for SQ to find x-intercept. The result is $\frac{29}{9}$.
### Final Answer
Correct Answer: Option C