Answer :
To solve the problem, let's go through it step by step:
1. Identify the Given Points:
- A(4, 6)
- B(2, 2)
- C(8, 2)
2. Find the Slope of Line BC:
- The formula for the slope of a line between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is [tex]\((y_2 - y_1) / (x_2 - x_1)\)[/tex].
- For BC, the coordinates are B(2, 2) and C(8, 2).
- Slope of BC: [tex]\((2 - 2) / (8 - 2) = 0/6 = 0\)[/tex].
- So, BC is a horizontal line.
3. Determine the Slope of AP:
- Since AP is perpendicular to BC, the slope of AP will be the negative reciprocal of the slope of BC.
- The slope of BC is 0, and the perpendicular to a horizontal line is a vertical line, which has an undefined slope. However, since AP must pass through point A, we can instead note that vertical lines are lines of constant x-values.
- Therefore, AP must be a vertical line passing through A, hence is [tex]\(x = 4\)[/tex].
4. Find the Co-ordinates of Point P:
- Since AP is vertical and BC is horizontal, the coordinates of point P would be where [tex]\(x = 4\)[/tex] (vertical line) intersects the line BC where [tex]\(y = 2\)[/tex] (since BC is horizontal).
- Therefore, point P has coordinates (4, 2).
Equation and Coordinates:
- Equation of AP: [tex]\(x = 4\)[/tex]
- Coordinates of Point P: [tex]\((4, 2)\)[/tex]
This gives us both the equation of the line AP and the coordinates of the point P where AP is perpendicular to BC.
1. Identify the Given Points:
- A(4, 6)
- B(2, 2)
- C(8, 2)
2. Find the Slope of Line BC:
- The formula for the slope of a line between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is [tex]\((y_2 - y_1) / (x_2 - x_1)\)[/tex].
- For BC, the coordinates are B(2, 2) and C(8, 2).
- Slope of BC: [tex]\((2 - 2) / (8 - 2) = 0/6 = 0\)[/tex].
- So, BC is a horizontal line.
3. Determine the Slope of AP:
- Since AP is perpendicular to BC, the slope of AP will be the negative reciprocal of the slope of BC.
- The slope of BC is 0, and the perpendicular to a horizontal line is a vertical line, which has an undefined slope. However, since AP must pass through point A, we can instead note that vertical lines are lines of constant x-values.
- Therefore, AP must be a vertical line passing through A, hence is [tex]\(x = 4\)[/tex].
4. Find the Co-ordinates of Point P:
- Since AP is vertical and BC is horizontal, the coordinates of point P would be where [tex]\(x = 4\)[/tex] (vertical line) intersects the line BC where [tex]\(y = 2\)[/tex] (since BC is horizontal).
- Therefore, point P has coordinates (4, 2).
Equation and Coordinates:
- Equation of AP: [tex]\(x = 4\)[/tex]
- Coordinates of Point P: [tex]\((4, 2)\)[/tex]
This gives us both the equation of the line AP and the coordinates of the point P where AP is perpendicular to BC.
