4.3 Q-2

Question Statement

In the triangle with vertices at $A(8, 6)$, $B(-4, 2)$, and $C(-2, -6)$, find the slope of:

i. Each side of the triangle

ii. Each median of the triangle

iii. Each altitude of the triangle

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Background and Explanation

In this problem, we are asked to calculate the slopes of the sides, medians, and altitudes of a triangle.

• The slope of a line is the ratio of the change in $y$ to the change in $x$ between two points on the line. It can be calculated using the formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

• A median of a triangle connects a vertex to the midpoint of the opposite side.
• An altitude of a triangle is a perpendicular line drawn from a vertex to the opposite side.

We will apply the slope formula to calculate the slopes of the sides, medians, and altitudes.

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Solution

i. Slope of Each Side of the Triangle

The vertices of the triangle are:
$A(8, 6)$, $B(-4, 2)$, and $C(-2, -6)$.

1. Slope of side $\overline{AB}$:

Using the slope formula:

$$m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 6}{-4 - 8} = \frac{-4}{-12} = \frac{1}{3}$$

2. Slope of side $\overline{BC}$:

Using the slope formula:

$$m_2 = \frac{-6 - 2}{-2 - (-4)} = \frac{-8}{2} = -4$$

3. Slope of side $\overline{AC}$:

Using the slope formula:

$$m_3 = \frac{-6 - 6}{-2 - 8} = \frac{-12}{-10} = \frac{6}{5}$$

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ii. Slope of Each Median

The midpoints of the sides are calculated as follows:

• Let $D$, $E$, and $F$ be the midpoints of sides $\overline{BC}$, $\overline{AC}$, and $\overline{AB}$, respectively.

1. Coordinates of midpoint $D$:

$$D = \left(\frac{-4 + 2}{2}, \frac{2 + 6}{2}\right) = (-3, -2)$$

2. Coordinates of midpoint $E$:

$$E = \left(\frac{8 + 2}{2}, \frac{6 + 6}{2}\right) = (3, 0)$$

3. Coordinates of midpoint $F$:

$$F = \left(\frac{8 + (-4)}{2}, \frac{6 + 2}{2}\right) = (2, 4)$$

Now, calculate the slopes of the medians:

1. Slope of median $\overline{AD}$:

Using the slope formula:

$$m_{\text{AD}} = \frac{-2 - 6}{-3 - 8} = \frac{-8}{-11} = \frac{8}{11}$$

2. Slope of median $\overline{BE}$:

Using the slope formula:

$$m_{\text{BE}} = \frac{0 - 2}{3 - (-4)} = \frac{-2}{7}$$

3. Slope of median $\overline{CF}$:

Using the slope formula:

$$m_{\text{CF}} = \frac{4 - (-6)}{4 - (-2)} = \frac{10}{6} = \frac{5}{3}$$

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iii. Slope of Each Altitude

The slope of an altitude is the negative reciprocal of the slope of the corresponding side, since altitudes are perpendicular to the sides.

1. Slope of the altitude through vertex $A$:

The slope of side $\overline{BC}$ is $-4$. The slope of the altitude through $A$ is the negative reciprocal of this:

$$m_{\text{altitude at A}} = \frac{-1}{-4} = \frac{1}{4}$$

2. Slope of the altitude through vertex $B$:

The slope of side $\overline{AC}$ is $\frac{6}{5}$. The slope of the altitude through $B$ is the negative reciprocal of this:

$$m_{\text{altitude at B}} = \frac{-1}{\frac{6}{5}} = \frac{-5}{6}$$

3. Slope of the altitude through vertex $C$:

The slope of side $\overline{AB}$ is $\frac{1}{3}$. The slope of the altitude through $C$ is the negative reciprocal of this:

$$m_{\text{altitude at C}} = \frac{-1}{\frac{1}{3}} = -3$$

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Key Formulas or Methods Used

• Slope formula:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

• Negative reciprocal:
  For perpendicular lines, if the slope of one line is $m$, the slope of the perpendicular line is $\frac{-1}{m}$.

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Summary of Steps

1. Calculate the slope of each side using the slope formula.
2. Find the midpoints of the sides to determine the coordinates of the medians.
3. Calculate the slope of each median using the slope formula.
4. Calculate the slope of each altitude by finding the negative reciprocal of the slope of the corresponding side.

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