Study the following diagram of circle C,
where AB¯¯¯¯¯¯¯¯
is a diameter that bisects the chord EF¯¯¯¯¯¯¯¯.
The circle as described in the problem. Segments C E and C F are marked congruent. If m∠ACF=2(7x−4),
what is the value of x?
Enter the correct value.
Since AE and BD are both radii of the circle, they must be congruent. Therefore, triangle ADC is isosceles.
This means that m∠CAD = m∠CDA
Since AB bisects EF, it must be perpendicular to EF. Therefore, m∠AFC = 90 degrees.
Since BF is a straight line, m∠BFA must be 90 degrees as well.
This means that m∠DCF = 180 - m∠DCF - m∠CAD = 180 - (m∠CDA + m∠CAD) = 180 - 2m∠CAD
Since the sum of angles in a triangle equals 180 degrees, we have the following equation:
m∠CFD + m∠CDF + 2m∠CAD = 180
Substitute in the given information:
(7x - 4) + 180 - 2(7x - 4) = 180
7x - 4 + 180 - 14x + 8 = 180
176 - 7x = 180
176 - 180 = 7x
-4 = 7x
x = -4/7
Therefore, the value of x is -4/7.