error analysis of secant angle
Check the final answer first, then review the worked steps.
Problem
error analysis of secant angle
Answer
36
Step-by-step solution
- Identify the geometric configuration: The problem involves two secant lines, $XZ$ and $XV$, intersecting at an external point $X$ outside the circle. The angle formed by two secants intersecting outside a circle is equal to half the difference of the measures of the intercepted arcs.
- State the correct theorem: For an angle formed by two secants intersecting outside a circle, the measure of the angle is given by the formula: $$m\angle X = \frac{1}{2}(m\text{arc}_{far} - m\text{arc}_{near})$$ In this specific case, the intercepted arcs are the far arc $VZ$ and the near arc $WY$. Therefore, the correct formula is: $$m\angle VXZ = \frac{1}{2}(m\widehat{VZ} - m\widehat{WY})$$
- Analyze Cindy's error: Cindy used the formula for an angle formed by two chords intersecting inside a circle, which is $m\angle = \frac{1}{2}(m\text{arc}_1 + m\text{arc}_2)$. She incorrectly added the measures of the arcs instead of subtracting them.
- Calculate the correct value: Using the correct formula with the given values $m\widehat{VZ} = 96^\circ$ and $m\widehat{WY} = 24^\circ$: $$m\angle VXZ = \frac{1}{2}(96^\circ - 24^\circ) = \frac{1}{2}(72^\circ) = 36^\circ$$