Hi! Can you please help me with this problem/proof? I'm not sure how to approach it. Thanks! Prove that a line perpendicular to a radius at the point where the radius meets the circle is tangent to the circle. Use coordinate geometry.Hi! Can you please help me with this problem/proof? I'm not sure how to approach it. Thanks! Prove that a line perpendicular to a radius at the point where the radius meets the circle is tangent to the circle. Use coordinate geometry.
Of course! I'd be happy to help you with this proof using coordinate geometry.
To prove that a line perpendicular to a radius at the point where the radius meets the circle is tangent to the circle, we can follow these steps:
1. Assume that the center of the circle is at the origin (0, 0) and the radius length is 'r'. Let's also consider the point where the radius meets the circle as (x, y).
2. Since the line is perpendicular to the radius, its slope will be the negative reciprocal of the slope of the radius. The slope of the radius line can be found using the formula: m = (y - 0) / (x - 0) = y / x.
3. Therefore, the slope of the line perpendicular to the radius is -x / y.
4. Now, let's consider a point on the line given by (a, b).
5. Since the line is perpendicular to the radius, the product of their slopes should be -1. So, we have (-x / y) * (b - y) / (a - x) = - 1.
6. By simplifying the equation, we get (b - y) * x + (a - x) * y = 0.
7. Expanding the equation, we have bx - xy + ay - xy = 0, which further simplifies to (a + b)x = (x + y)(x - y).
8. Since (a + b) is a constant value and (x + y)(x - y) = x^2 - y^2, we can rewrite the equation as x^2 - y^2 = 0.
9. This equation represents a circle with radius 'r'. If we substitute x = r and y = 0 in this equation, we have r^2 - 0 = 0, which is a true statement.
10. Therefore, the given line intersects the circle at its center, making it tangent to the circle.
By following these steps, we have proved that a line perpendicular to a radius at the point where the radius meets the circle is indeed tangent to the circle.