4.3 Q-3

Question Statement

Given the points:

a. $(-1, -3)$, $(1, 5)$, $(2, 9)$
b. $(4, -5)$, $(7, 5)$, $(10, 15)$
c. $(a, 2b)$, $(c, a+b)$, $(2c-a, 2a)$

Use the concept of slopes to show that these points lie on the same straight line (i.e., they are collinear).

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Background and Explanation

To determine if three points lie on the same straight line, we use the concept of slopes. The slope of a line between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula:

$$\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}$$

If the slopes between any two pairs of points are equal, then the points are collinear. We will calculate the slope between different pairs of points and check if they are equal.

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Solution

a. Points $(-1, -3)$, $(1, 5)$, $(2, 9)$

1. Let the points be $A(-1, -3)$, $B(1, 5)$, and $C(2, 9)$.
2. Slope of line $\overline{AB}$:
   Using the formula for the slope:

$$m_1 = \frac{5 - (-3)}{1 - (-1)} = \frac{5 + 3}{2} = \frac{8}{2} = 4$$

3. Slope of line $\overline{BC}$:

$$m_2 = \frac{9 - 5}{2 - 1} = \frac{4}{1} = 4$$

4. Slope of line $\overline{AC}$:

$$m_3 = \frac{9 - (-3)}{2 - (-1)} = \frac{9 + 3}{3} = \frac{12}{3} = 4$$

Since all the slopes are equal, we can conclude that the points $A$, $B$, and $C$ are collinear.

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b. Points $(4, -5)$, $(7, 5)$, $(10, 15)$

1. Let the points be $A(4, -5)$, $B(7, 5)$, and $C(10, 15)$.
2. Slope of line $\overline{AB}$:

$$m_1 = \frac{5 - (-5)}{7 - 4} = \frac{5 + 5}{3} = \frac{10}{3}$$

3. Slope of line $\overline{BC}$:

$$m_2 = \frac{15 - 5}{10 - 7} = \frac{10}{3}$$

4. Slope of line $\overline{AC}$:

$$m_3 = \frac{15 - (-5)}{10 - 4} = \frac{15 + 5}{6} = \frac{20}{6} = \frac{10}{3}$$

Since all the slopes are equal, the points $A$, $B$, and $C$ are collinear.

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c. Points $(a, 2b)$, $(c, a+b)$, $(2c-a, 2a)$

1. Let the points be $A(a, 2b)$, $B(c, a+b)$, and $C(2c-a, 2a)$.
2. Slope of line $\overline{AB}$:

$$m_1 = \frac{(a + b) - 2b}{c - a} = \frac{a - b}{c - a}$$

3. Slope of line $\overline{BC}$:

$$m_2 = \frac{2a - (a + b)}{2c - a - c} = \frac{2a - a - b}{c - a} = \frac{a - b}{c - a}$$

4. Slope of line $\overline{AC}$:

$$m_3 = \frac{2a - 2b}{2c - a - a} = \frac{2(a - b)}{2(c - a)} = \frac{a - b}{c - a}$$

Since all the slopes are equal, we can conclude that the points $A$, $B$, and $C$ are collinear.

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Key Formulas or Methods Used

• Slope Formula:

$$\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}$$

• Condition for Collinearity:
  If the slopes between any two pairs of points are equal, then the points are collinear.

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Summary of Steps

1. Calculate the slope of each line formed by two points.
2. Compare the slopes of the lines.
3. If all slopes are equal, the points are collinear.