Chapter 47
Angle of Intersecting Chords Theorem Concept Explanation:
The Angle of Intersecting Chords Theorem states that if two chords intersect inside a circle, the measure of the angle formed by these intersecting chords is half the sum of the measures of the arcs intercepted by the angle and its vertical angle. In other words, if two chords and intersect at a point inside a circle, then the measure of is given by:
This theorem is useful for calculating angles formed by intersecting chords in circles.
*** QuickLaTeX cannot compile formula: \begin{tikzpicture} \draw (0,0) circle (3cm); \draw<a href="-2,2">thick</a> -- (2,-2); \draw<a href="-2,-2">thick</a> -- (2,2); \node at (-2,2) [left] {A}; \node at (2,-2) [right] {B}; \node at (-2,-2) [left] {C}; \node at (2,2) [right] {D}; \node at (0,0) [below] {E}; \end{tikzpicture} *** Error message: Error: Cannot create svg file
In this diagram, and are intersecting chords, and is their point of intersection.
Common Mistakes:
- Confusing the Sum of Arcs: Students sometimes mistakenly use the arc opposite the angle instead of the intercepted arcs.
- Forgetting to Halve the Sum: After calculating the sum of the arcs, it’s common to forget dividing by 2.
- Misinterpreting Vertical Angles: Vertical angles formed by intersecting chords are congruent, and this can be overlooked.
Helpful Tips:
- Draw Arcs Clearly: Always mark the intercepted arcs to avoid confusion.
- Check Vertical Angles: If needed, use the fact that vertical angles are equal when chords intersect.
Hard Questions:
Q1: In a circle, two chords and intersect at point . The measure of is 80°, and the measure of is 60°. Find .
Step-by-Step Solution:
- Apply the Angle of Intersecting Chords Theorem:
Substitute the known values:
Answer:
Q2: If the intercepted arcs in a circle are and , and the chords intersect at point , what is the measure of ?
Step-by-Step Solution:
- Use the Angle of Intersecting Chords Theorem:
Substitute the given arc measures:
Answer:
Q3: In a circle, two chords intersect, and the intercepted arcs are and . Find the measure of the angle formed by the intersecting chords.
Step-by-Step Solution:
- Use the Angle of Intersecting Chords Theorem:
Substitute the known values:
Answer:
Q4: Two chords and intersect at point inside a circle. The intercepted arcs and are 90° and 70° respectively. What is ?
Step-by-Step Solution:
- Apply the Angle of Intersecting Chords Theorem:
Substitute the arc measures:
Answer:
Q5: If two chords intersect inside a circle such that and , find .
Step-by-Step Solution:
- Use the Angle of Intersecting Chords Theorem:
Substitute the values:
Answer: