Question

question 1 (multiple choice worth 1 points)
(07.01 mc)
find the measure of (overparen{eb}).
(25^{circ})
(96^{circ})
(118^{circ})
(146^{circ})

Explanation:

Step1: Use the formula for the angle formed by a tangent and a secant

The formula is \(\angle C=\frac{1}{2}(\text{measure of arc }DB - \text{measure of arc }EB)\). We know that \(\angle C = 25^{\circ}\), arc \(DB=(7x + 6)^{\circ}\) and arc \(EB=(4x + 16)^{\circ}\). So, \(25=\frac{1}{2}((7x + 6)-(4x + 16))\).

Step2: Solve the equation for \(x\)

First, multiply both sides of the equation \(25=\frac{1}{2}((7x + 6)-(4x + 16))\) by \(2\): \(50=(7x + 6)-(4x + 16)\).
Then, simplify the right - hand side: \(50=7x + 6-4x-16\).
Combine like terms: \(50 = 3x-10\).
Add \(10\) to both sides: \(50+10=3x\), so \(60 = 3x\).
Divide both sides by \(3\): \(x = 20\).

Step3: Find the measure of arc \(EB\)

Substitute \(x = 20\) into the expression for arc \(EB\). Arc \(EB=(4x + 16)^{\circ}\).
\(EB=(4\times20 + 16)^{\circ}=(80 + 16)^{\circ}=96^{\circ}\).

Answer:

\(96^{\circ}\)