Find Measure of Arc FH: Intersecting Secants Theorem

Answer

The measure of the minor arc $\widehat{FH}$ is found by applying the Intersecting Secants Theorem, which relates the exterior angle to the intercepted arcs. By solving for $x$ and substituting it back into the expression for the arc, we find that $m\widehat{FH} = 70^{\circ}$.

Explanation

In the provided image, we observe two secant lines, $\overline{GC}$ and $\overline{GE}$, intersecting at an external point $G$. These secants intercept two arcs on the circle: the major arc $\widehat{CE}$ and the minor arc $\widehat{FH}$.

1. Identify Known Quantities From the diagram, we can extract the following algebraic and numerical values for the geometric components:

   • Measure of the exterior angle: $m\angle G = 15x$
   • Measure of the major intercepted arc: $m\widehat{CE} = 130^{\circ}$
   • Measure of the minor intercepted arc: $m\widehat{FH} = 36x - 2$

2. Formula Selection According to the Intersecting Secants Theorem (or the Exterior Angle of a Circle Theorem), the measure of an angle formed by two secants intersecting outside a circle is half the difference of the measures of the intercepted arcs. $m\angle G = \frac{1}{2}(m\widehat{CE} - m\widehat{FH})$ This formula states that the external angle is equal to one-half the major arc minus the minor arc. ⚠️ This step is required on exams to justify your equation.

3. Substitution and Algebraic Solving Substitute the given expressions into the theorem's formula to solve for the variable $x$: $15x = \frac{1}{2}(130 - (36x - 2))$ Multiply both sides by 2 to eliminate the fraction: $30x = 130 - 36x + 2$ This simplifies the equation into a linear form for easier manipulation. Combine constant terms on the right side: $30x = 132 - 36x$ Add $36x$ to both sides to isolate the $x$ terms: $66x = 132$ Divide by 66: $x = 2$ This value represents the scale factor for our geometric expressions.

4. Calculate the Measure of Arc FH Now, substitute the value $x = 2$ back into the expression provided for arc $\widehat{FH}$: $m\widehat{FH} = 36(2) - 2$ $m\widehat{FH} = 72 - 2$ $m\widehat{FH} = 70$ The numerical value represents the degree measure of the arc length relative to the circle's center.

5. Unit Check In circle geometry, arc measures and angles are typically measured in degrees ($^{\circ}$). Since $130 - 70 = 60$, and half of 60 is 30, we can verify that $15(2) = 30$. The units and logic are consistent.

Final Answer

The measure of arc $\widehat{FH}$ is: $\boxed{70^{\circ}}$

Common Mistakes

• Sign Errors: Students often forget to distribute the negative sign when subtracting the minor arc expression, writing $130 - 36x - 2$ instead of $130 - 36x + 2$.
• Misidentifying Arcs: Using the sum of the arcs instead of the difference; addition is only used when the lines intersect inside the circle (Chord-Chord Power Theorem).