Question

question
solve for x.
answer attempt 1 out of 3
additional solution no solution

Explanation:

Step1: Use the tangent - secant angle formula

The measure of an angle formed by a tangent and a secant is half the difference of the measures of the intercepted arcs. The formula is \(\angle CDF=\frac{1}{2}(m\overset{\frown}{CF}-m\overset{\frown}{CE})\). Here, \(\angle CDF=(21x - 3)^{\circ}\), \(m\overset{\frown}{CF}=(67x-1)^{\circ}\), and \(m\overset{\frown}{CE} = 80^{\circ}\). So, \(21x-3=\frac{1}{2}[(67x - 1)-80]\).

Step2: Simplify the right - hand side

First, simplify the expression inside the brackets: \((67x-1)-80=67x-(1 + 80)=67x-81\). Then, \(\frac{1}{2}(67x - 81)=\frac{67x}{2}-\frac{81}{2}\). So the equation becomes \(21x-3=\frac{67x}{2}-\frac{81}{2}\).

Step3: Eliminate the fraction

Multiply through by 2 to get rid of the denominators: \(2(21x-3)=67x-81\). Expand the left - hand side: \(42x-6 = 67x-81\).

Step4: Solve for \(x\)

Subtract \(42x\) from both sides: \(-6=67x-42x-81\). Simplify the right - hand side: \(-6 = 25x-81\). Add 81 to both sides: \(25x=-6 + 81\). So, \(25x=75\). Divide both sides by 25: \(x = 3\).

Answer:

\(x = 3\)