4.3 Q-15

Question Statement

Given points $A(-1, 2)$, $B(6, 3)$, and $C(2, -4)$ as the vertices of a triangle, show that the line joining the mid-points $D$ of $AB$ and $E$ of $AC$ is parallel to $BC$, and that $DE = \frac{1}{2} BC$.

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Background and Explanation

To solve this problem, we need:

1. Midpoint Formula: The midpoint of a line segment joining two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:
   $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

2. Slope Formula: The slope of a line joining two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:
   $\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}$
   Parallel lines have the same slope.

3. Distance Formula: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:
   $\text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

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Solution

1. Find the Midpoints $D$ and $E$:

   • Midpoint $D$ of $AB$:
     $D = \left( \frac{-1 + 6}{2}, \frac{2 + 3}{2} \right) = D \left( \frac{5}{2}, \frac{5}{2} \right)$
   • Midpoint $E$ of $AC$:
     $E = \left( \frac{-1 + 2}{2}, \frac{2 - 4}{2} \right) = E \left( \frac{1}{2}, -1 \right)$

2. Find the Slopes of $DE$ and $BC$:

   • Slope of $DE$:
     $\text{slope of } DE = \frac{-1 - \frac{5}{2}}{\frac{1}{2} - \frac{5}{2}} = \frac{-\frac{7}{2}}{-\frac{4}{2}} = \frac{7}{4}$
   • Slope of $BC$:
     $\text{slope of } BC = \frac{-4 - 3}{2 - 6} = \frac{-7}{-4} = \frac{7}{4}$

   Since the slopes of $DE$ and $BC$ are equal, we conclude that:
   $\overline{DE} \parallel \overline{BC}$

3. Verify that $DE = \frac{1}{2} BC$:

   • Distance $DE$:
     $\mid \overline{DE} = \sqrt{\left( \frac{1}{2} - \frac{5}{2} \right)^2 + \left( -1 - \frac{5}{2} \right)^2} = \sqrt{\left( \frac{-4}{2} \right)^2 + \left( \frac{-7}{2} \right)^2}$
     $= \sqrt{\frac{16}{4} + \frac{49}{4}} = \sqrt{\frac{65}{4}} = \frac{\sqrt{65}}{2}$
   • Distance $BC$:
     $\mid \overline{BC} = \sqrt{(2 - 6)^2 + (-4 - 3)^2} = \sqrt{16 + 49} = \sqrt{65}$

   Since $DE = \frac{1}{2} BC$, we conclude:
   $D E = \frac{1}{2} B C$

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Key Formulas or Methods Used

• Midpoint Formula:
  $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$

• Slope Formula:
  $\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}$
  Used to determine parallelism.

• Distance Formula:
  $\text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
  Used to check if $DE = \frac{1}{2} BC$.

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Summary of Steps

1. Find the midpoints $D$ and $E$ using the midpoint formula.
2. Find the slopes of $DE$ and $BC$ using the slope formula.
3. Check parallelism by comparing the slopes of $DE$ and $BC$.
4. Calculate the distances $DE$ and $BC$ and verify that $DE = \frac{1}{2} BC$.