Question

in the figure below, \\(\overline{ab}\\) is a diameter of circle \\(p\\). what is the arc measure of minor arc \\(\widehat{ad}\\) in degrees? circle with center p, diameter ab, angles at p: \\((7x + 1)\degree\\) for \\(\angle apd\\), right angle between pd and pc, \\((9x - 7)\degree\\) for \\(\angle cpb\\)

Explanation:

Step1: Identify the straight angle

Since \( AB \) is a diameter, \( \angle APB = 180^\circ \). The angles \( \angle APD=(7x + 1)^\circ \), the right angle (let's assume the middle angle is \( 90^\circ \) as it looks like a right angle in the diagram), and \( \angle CPB=(9x - 7)^\circ \) add up to \( 180^\circ \). So the equation is:

$$ (7x + 1)+90+(9x - 7)=180 $$

Step2: Solve for \( x \)

Combine like terms:

$$ 7x+9x + 1-7 + 90=180 $$

$$ 16x+84 = 180 $$

Subtract 84 from both sides:

$$ 16x=180 - 84 $$

$$ 16x = 96 $$

Divide both sides by 16:

$$ x=\frac{96}{16}=6 $$

Step3: Find the measure of \( \angle APD \)

Substitute \( x = 6 \) into \( (7x + 1)^\circ \):

$$ 7(6)+1=42 + 1=43^\circ $$

The measure of minor arc \( \widehat{AD} \) is equal to the measure of its central angle \( \angle APD \), so the arc measure is \( 43^\circ \).

Answer:

\( 43 \)