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Can anybody help me with this question please?

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- Jun 23rd 2007, 11:23 AMGAdamsGCSE Question
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Can anybody help me with this question please? - Jun 23rd 2007, 01:11 PMJhevon
First let's find the point M

The formula for the midpoint is given by:

$\displaystyle \left( \frac {x_1 + x_2}{2}, \frac {y_1 + y_2}{2} \right)$

Using $\displaystyle (x_1,y_1) = A(2,9)$ and $\displaystyle (x_2, y_2) = B(8,7)$, the point M is:

$\displaystyle M = \left( \frac {2 + 8}{2}, \frac {9 + 7}{2} \right) = (5,8)$

Now we can find the slope of the lines connecting M and C, and the slope of the line connecting A and B, if there slopes are the negative inverses of each other (that is, the product of their slopes is -1), then MC is perpendicular to AB.

Let $\displaystyle m_1$ be the slope of the line connecting AB

Let $\displaystyle m_2$ be the slope of the line connecting MC

Now, using $\displaystyle (x_1,y_1) = (2,9) \mbox { and } (x_2,y_2) = (8,7)$

$\displaystyle \Rightarrow m_1 = \frac {y_2 - y_1}{x_2 - x_1} = \frac {7-9}{8-2} = - \frac {1}{3}$

Now, using $\displaystyle (x_1,y_1) = (5,8) \mbox { and } (x_2 , y_2) = (8,18)$

$\displaystyle \Rightarrow m_2 = \frac {18 - 8}{8 - 5} = \frac {10}{3}$

Thus, MC and AB are not perpendicular