The problem provides a circle with center $D$. $PM$ and $NL$ are chords of the circle. The length of $PD$ and $DL$ is 7. The measure of arc $MP$ is given by $(4x + 22)^\circ$, and the measure of arc $NL$ is given by $(5x + 8)^\circ$. (a) We need to find the value of $x$. (b) We need to find the measure of arc $MP$.

1. Problem Description

The problem provides a circle with center $D$. $PM$ and $NL$ are chords of the circle. The length of $PD$ and $DL$ is

7. The measure of arc $MP$ is given by $(4x + 22)^\circ$, and the measure of arc $NL$ is given by $(5x + 8)^\circ$.

(a) We need to find the value of $x$.

(b) We need to find the measure of arc $MP$.

2. Solution Steps

(a) Since $PM$ and $NL$ are chords and the distance from the center of the circle to the chords is the same ($PD = DL = 7$), then the chords are equidistant from the center, which implies that they are congruent. If the chords are congruent, then the arcs they intercept are also congruent. Therefore, we can write:

$m\widehat{MP} = m\widehat{NL}$

$4x + 22 = 5x + 8$

Subtract $4x$ from both sides:

$22 = x + 8$

Subtract 8 from both sides:

$x = 14$

(b) Now that we have found the value of $x$, we can find the measure of arc $MP$:

$m\widehat{MP} = (4x + 22)^\circ$

Substitute $x = 14$:

$m\widehat{MP} = (4(14) + 22)^\circ$

$m\widehat{MP} = (56 + 22)^\circ$

$m\widehat{MP} = 78^\circ$

3. Final Answer

x = 14

$m\widehat{MP} = 78^\circ$