Answer :
To solve this problem, follow these steps closely:
### Step 1: Identify the Coordinates of Points A, B, and C
- Point A has coordinates (1, 1)
- Point B has coordinates (2, 6)
- Point C has coordinates (4, 2)
### Step 2: Calculate the Midpoint [tex]\(M\)[/tex] of Line Segment BC
The midpoint M of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be calculated using the midpoint formula:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Applying this formula to points B and C:
[tex]\[ M = \left( \frac{2 + 4}{2}, \frac{6 + 2}{2} \right) \][/tex]
[tex]\[ M = (3.0, 4.0) \][/tex]
### Step 3: Determine the Slope of Line AM
The slope [tex]\( m \)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using point A (1, 1) and the midpoint M (3.0, 4.0):
[tex]\[ m = \frac{4.0 - 1}{3.0 - 1} \][/tex]
[tex]\[ m = \frac{3.0}{2.0} \][/tex]
[tex]\[ m = 1.5 \][/tex]
### Step 4: Use the Point-Slope Form to Calculate the Y-Intercept
The point-slope form of the equation of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
We can use point A (1, 1) and the slope [tex]\( m = 1.5 \)[/tex]:
[tex]\[ y - 1 = 1.5(x - 1) \][/tex]
Expanding and solving for [tex]\( y \)[/tex]:
[tex]\[ y - 1 = 1.5x - 1.5 \][/tex]
[tex]\[ y = 1.5x - 1.5 + 1 \][/tex]
[tex]\[ y = 1.5x - 0.5 \][/tex]
Therefore, the y-intercept [tex]\( b \)[/tex]:
[tex]\[ b = -0.5 \][/tex]
### Final Equation of the Line
Putting it all together, the equation of the line passing through points A and the midpoint M is:
[tex]\[ y = 1.5x - 0.5 \][/tex]
So, the final answer is:
- Midpoint [tex]\(M\)[/tex]: (3.0, 4.0)
- Slope: 1.5
- Y-intercept: -0.5
- Equation: [tex]\( y = 1.5x - 0.5 \)[/tex]
### Step 1: Identify the Coordinates of Points A, B, and C
- Point A has coordinates (1, 1)
- Point B has coordinates (2, 6)
- Point C has coordinates (4, 2)
### Step 2: Calculate the Midpoint [tex]\(M\)[/tex] of Line Segment BC
The midpoint M of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be calculated using the midpoint formula:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Applying this formula to points B and C:
[tex]\[ M = \left( \frac{2 + 4}{2}, \frac{6 + 2}{2} \right) \][/tex]
[tex]\[ M = (3.0, 4.0) \][/tex]
### Step 3: Determine the Slope of Line AM
The slope [tex]\( m \)[/tex] of a line passing through two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Using point A (1, 1) and the midpoint M (3.0, 4.0):
[tex]\[ m = \frac{4.0 - 1}{3.0 - 1} \][/tex]
[tex]\[ m = \frac{3.0}{2.0} \][/tex]
[tex]\[ m = 1.5 \][/tex]
### Step 4: Use the Point-Slope Form to Calculate the Y-Intercept
The point-slope form of the equation of a line is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
We can use point A (1, 1) and the slope [tex]\( m = 1.5 \)[/tex]:
[tex]\[ y - 1 = 1.5(x - 1) \][/tex]
Expanding and solving for [tex]\( y \)[/tex]:
[tex]\[ y - 1 = 1.5x - 1.5 \][/tex]
[tex]\[ y = 1.5x - 1.5 + 1 \][/tex]
[tex]\[ y = 1.5x - 0.5 \][/tex]
Therefore, the y-intercept [tex]\( b \)[/tex]:
[tex]\[ b = -0.5 \][/tex]
### Final Equation of the Line
Putting it all together, the equation of the line passing through points A and the midpoint M is:
[tex]\[ y = 1.5x - 0.5 \][/tex]
So, the final answer is:
- Midpoint [tex]\(M\)[/tex]: (3.0, 4.0)
- Slope: 1.5
- Y-intercept: -0.5
- Equation: [tex]\( y = 1.5x - 0.5 \)[/tex]
