| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2007 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Straight Lines & Coordinate Geometry |
| Type | Perpendicular bisector of segment |
| Difficulty | Moderate -0.3 This is a straightforward coordinate geometry question requiring standard techniques: finding a midpoint (formula recall), calculating gradient and perpendicular gradient, verifying a line equation, then finding a triangle area. Part (ii) requires finding intersection points and applying the triangle area formula. While multi-step, all techniques are routine C1 content with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.03a Straight lines: equation forms y=mx+c, ax+by+c=01.03b Straight lines: parallel and perpendicular relationships |
## Question 12
### (i) Find the coordinates of the midpoint, M, of AB. Show also that the equation of the perpendicular bisector of AB is $y = 2x + 12$. [6]
M1: Find midpoint using $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
A1: Midpoint coordinates are $(3, 6)$
M1: Find gradient of AB using $\frac{y_2 - y_1}{x_2 - x_1}$
A1: Gradient of AB is $\frac{1}{2}$
M1: Find gradient of perpendicular bisector (negative reciprocal)
A1: Gradient of perpendicular bisector is $-2$
M1: Use point-slope form $y - y_1 = m(x - x_1)$ with point M and gradient $-2$
A1: Equation of perpendicular bisector is $y = -2x + 12$ or equivalent
### (ii) Find the area of the triangle bounded by the perpendicular bisector, the y-axis and the line AM. [6]
M1: Find equation of line AM using points A and M
A1: Equation of line AM is $y = 4$ (or equivalent working showing gradient is 0)
M1: Find intersection of perpendicular bisector and y-axis (set $x = 0$)
A1: Point is $(0, 12)$
M1: Find intersection of line AM and y-axis (set $x = 0$)
A1: Point is $(0, 4)$
M1: Find intersection of perpendicular bisector and line AM
A1: Point is $(4, 4)$
M1: Use formula for area of triangle, e.g., $\frac{1}{2} \times \text{base} \times \text{height}$
A1: Area is $16$ square units (or equivalent)
---12 Use coordinate geometry to answer this question. Answers obtained from accurate drawing will receive no marks.\\
$A$ and $B$ are points with coordinates $( - 1,4 )$ and $( 7,8 )$ respectively.\\
(i) Find the coordinates of the midpoint, M , of AB .
Show also that the equation of the perpendicular bisector of AB is $y + 2 x = 12$.\\
(ii) Find the area of the triangle bounded by the perpendicular bisector, the $y$-axis and the line AM , as sketched in Fig. 12.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{7791371e-26d9-428c-8700-5121a1c6464a-4_451_483_776_790}
\captionsetup{labelformat=empty}
\caption{Fig. 12}
\end{center}
\end{figure}
Not to scale
\hfill \mbox{\textit{OCR MEI C1 2007 Q12 [12]}}