Question

in the circle below, if $overline{pq}=overline{qr}$, $overline{jk}=3x + 23$ and $overline{lm}=9x - 19$, find $overline{pk}$. the length of $overline{pk}=$ options: 7, 11, 22, 44 select one: a. 58° b. 84° c. 34° d. 29° in the diagram below, if $mangle bac = 29^{circ}$, what is the $mangle bdc$?

Explanation:

Step1: Use property of chords equidistant from center

If two chords are equidistant from the center of a circle, they are congruent. Since $PQ = QR$, chords $JK$ and $LM$ are congruent. So, $JK=LM$.
$3x + 23=9x - 19$

Step2: Solve for $x$

Subtract $3x$ from both sides:
$23 = 9x-3x - 19$
$23=6x - 19$
Add 19 to both sides:
$23 + 19=6x$
$42 = 6x$
Divide both sides by 6:
$x = 7$

Step3: Find length of $JK$

Substitute $x = 7$ into the expression for $JK$:
$JK=3x + 23=3\times7+23=21 + 23=44$
Since $PK$ is half of $JK$ (a perpendicular from the center to a chord bisects the chord), and $JK = 44$, then $PK = 22$

For the angle - related part:
The measure of an inscribed angle is half of the measure of the intercepted arc. Angles $\angle BAC$ and $\angle BDC$ intercept the same arc $\overset{\frown}{BC}$. So, $m\angle BDC=m\angle BAC = 29^{\circ}$

Answer:

The length of $PK$ is 22. The measure of $\angle BDC$ is $29^{\circ}$