Moderate -0.3 This is a straightforward multi-part coordinate geometry question requiring standard techniques: finding a midpoint (routine formula), verifying a perpendicular bisector equation (gradient calculation and point-slope form), and calculating a triangle area. While it has multiple steps, each component is a textbook exercise with no novel insight required, making it slightly easier than average.
1 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.
Find the coordinates of the midpoint, M , of AB .
Show also that the equation of the perpendicular bisector of AB is \(y + 2 x = 12\).
Find the area of the triangle bounded by the perpendicular bisector, the \(y\)-axis and the line AM , as sketched in Fig. 12.
1 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{tikzpicture}[>=stealth, scale=0.5]
% Shaded triangle: A(-1,4) -- M(3,6) -- (0,12)
\filldraw[gray!60] (0, 4.5) -- (3,6) -- (0, 12) -- cycle;
% Axes
\draw[->] (-3,0) -- (9,0) node[right] {$x$};
\draw[->] (0,-1) -- (0,14) node[above] {$y$};
\node[below left] at (0,0) {$0$};
% Line AB: slope 1/2, y = x/2 + 9/2, extended beyond A and B
\draw (-3,3) -- (9,9);
% Line y + 2x = 12, i.e. y = -2x + 12, through M(3,6)
\draw (-0.5,13) -- (6.5,-1);
% Points
\filldraw[black] (-1,4) circle (3pt);
\node[above left] at (-1,4) {A};
\node[below left] at (-1,4) {$(-1,4)$};
\filldraw[black] (7,8) circle (3pt);
\node[above left] at (7,8) {B};
\node[below right] at (7,8) {$(7,8)$};
\filldraw[black] (3,6) circle (3pt);
\node[below right] at (3,6) {M};
\end{tikzpicture}
\hfill \mbox{\textit{OCR MEI C1 Q1 [12]}}