To solve for the coordinates of point C on the y-axis such that angle ABC is 90°, follow these steps:
1. Understand Point Placement:
- Point A is given as (0, 8).
- Point B is given as (8, 6).
- Point C lies on the y-axis, meaning its x-coordinate is 0, so assume C is (0, k).
2. Calculate the Slope of Line Segment AB:
- The formula for the slope 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].
- Thus, the slope of line AB is [tex]\((6 - 8)/(8 - 0) = -2/8 = -1/4\)[/tex].
3. Define the Slope of Line Segment BC:
- The slope of line BC will be [tex]\((k - 6) / (0 - 8) = (k - 6) / -8\)[/tex].
4. Utilize the Perpendicular Condition:
- For lines AB and BC to be perpendicular, their slopes must multiply to -1.
- Let [tex]\(m_1\)[/tex] be the slope of AB and [tex]\(m_2\)[/tex] be the slope of BC.
- Thus, [tex]\((-1/4) (k - 6) / -8 = -1\)[/tex].
5. Simplify and Solve the Equation:
- First, multiply out: [tex]\((-1/4) (k - 6)/-8 = -1\)[/tex]
- Simplify: [tex]\(1/4 (k - 6) / 8 = -1\)[/tex]
- Further simplify: [tex]\(1/32 (k - 6) = -1\)[/tex]
- Clear the fraction by multiplying both sides by 32: [tex]\(k - 6 = -32\)[/tex]
- Solve for [tex]\(k\)[/tex]: [tex]\(k = -32 + 6 = -26\)[/tex]
6. Result:
- The point C on the y-axis, where the angle ABC is 90°, is [tex]\((0, k)\)[/tex].
- Therefore, C is [tex]\((0, -26)\)[/tex].
So, the coordinates of point C are [tex]\((0, -26)\)[/tex]. This guarantees that the angle ABC is a right angle (90 degrees).
