Theorem 1 states that if a radius of a circle is perpendicular to a chord (a line segment with both endpoints on the circle), then that radius bisects, or divides, the chord into two equal parts.

Here's a step-by-step explanation of the theorem:

1. Circle and Radius Basics: A radius of a circle is a line segment that connects the center of the circle to a point on its circumference. In this theorem, we have a radius that meets a chord (a line segment within the circle that does not pass through the center).

2. Perpendicular Intersection: The theorem specifies that the radius is perpendicular to the chord, meaning they meet at a 90° angle. This perpendicularity is crucial because it leads to the bisection property.

3. Bisecting Property: Because the radius is perpendicular to the chord, it will divide the chord into two equal halves. In other words, the point where the radius intersects the chord is the midpoint of the chord. Thus, each half of the chord is of equal length.

Why This is True (Geometric Reasoning)

This property is based on the symmetry of the circle:

• When a radius is drawn perpendicular to a chord, it creates two right triangles on either side of the radius.
• Since the radius is the same distance to both points where the chord meets the circle, each right triangle has two sides of equal length, making them congruent by the Pythagorean theorem.
• As a result, the radius intersects the chord precisely at its midpoint, bisecting it.

This theorem is often used in geometry to help find lengths of chords, distances, or to prove further properties of circles.

Would you like further details or have any questions?

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Related Questions

1. How does this theorem apply to a circle's diameter and any chord it meets?
2. What are examples of problems where this theorem is essential?
3. Can this theorem help in determining the area of shapes involving circles?
4. Why is the perpendicular condition necessary for the bisection to occur?
5. How does this theorem relate to the concept of congruent triangles?

Tip

Always look for symmetry when working with circles—it often leads to key insights and simplifications in geometry problems!