Question

1. if $jg = jf$, $gd = 13$, and $m\widehat{cd} = 136^\circ$, find each measure.

$ed = \underline{\quad\quad}$
$cf = \underline{\quad\quad}$
$m\widehat{ed} = \underline{\quad\quad}$
$m\widehat{hd} = \underline{\quad\quad}$
$m\widehat{ce} = \underline{\quad\quad}$

1. if $yu = yv$, $st = 16$, $m\widehat{qs} = 34^\circ$, and $m\widehat{rt} = 98^\circ$, find each measure.

$qu = \underline{\quad\quad}$
$qr= \underline{\quad\quad}$
$m\widehat{st} = \underline{\quad\quad}$
$m\widehat{qr} = \underline{\quad\quad}$
$m\widehat{xt} = \underline{\quad\quad}$

1. if $pq = qr$, $jk = 3x + 23$ and $lm = 9x - 19$, find $pk$.
2. if $dh = he$, $m\widehat{bg} = (9x - 20)^\circ$ and $m\widehat{gc} = (5x + 28)^\circ$, find $m\widehat{ab}$.

use the circle below for questions 13 - 16.

1. find $nk$.
2. find $m\widehat{mk}$.
3. find $jk$.
4. find $m\widehat{jpk}$.

Explanation:

Problem 9

Step1: Find length of ED

Since $JG=JF$, radii $JI$ and $JH$ are congruent, so chords $ED$ and $CD$ are congruent. $GD=13$, so $ED=2\times GD=26$.

Step2: Find length of CF

Radius $JH\perp CD$, so it bisects $CD$. $m\overset{\frown}{CD}=136^\circ$, so $CF=\frac{1}{2}CD=13$.

Step3: Find $m\overset{\frown}{ED}$

Congruent chords have congruent arcs, so $m\overset{\frown}{ED}=m\overset{\frown}{CD}=136^\circ$.

Step4: Find $m\overset{\frown}{HD}$

Radius bisects the arc, so $m\overset{\frown}{HD}=\frac{1}{2}m\overset{\frown}{CD}=68^\circ$.

Step5: Find $m\overset{\frown}{CE}$

Total circle is $360^\circ$, so $m\overset{\frown}{CE}=360^\circ - 136^\circ -136^\circ=88^\circ$.

Problem 10

Step1: Find length of QU

$YU=YV$, radii $YW$ and $YX$ are congruent, so chords $QS$ and $ST$ are congruent. $ST=16$, so $QU=\frac{1}{2}QS=8$.

Step2: Find length of QR

$QR$ is a chord, $YU\perp QR$, $m\overset{\frown}{QS}=34^\circ$, $QR$ is congruent to $ST$? No, $YU=YV$ means $QR=ST=16$.

Step3: Find $m\overset{\frown}{ST}$

Congruent chords have congruent arcs, so $m\overset{\frown}{ST}=m\overset{\frown}{QS}=34^\circ$.

Step4: Find $m\overset{\frown}{QR}$

Total circle minus known arcs: $m\overset{\frown}{QR}=180^\circ - 34^\circ=146^\circ$ (since $WRXT$ is diameter? No, $m\overset{\frown}{RT}=98^\circ$, so $m\overset{\frown}{QR}=180^\circ - 98^\circ=82^\circ$? Wait, no: $m\overset{\frown}{QS}=34^\circ$, $m\overset{\frown}{RT}=98^\circ$, so $m\overset{\frown}{QR}=180^\circ - 34^\circ=146^\circ$ is wrong. Correct: $m\overset{\frown}{QR}=180^\circ - m\overset{\frown}{RT}=180^\circ-98^\circ=82^\circ$.

Step5: Find $m\overset{\frown}{XT}$

Radius bisects the arc, so $m\overset{\frown}{XT}=\frac{1}{2}m\overset{\frown}{ST}=17^\circ$.

Problem 11

Step1: Set JK=LM

Congruent chords have congruent distances from center, so $3x+23=9x-19$.

Step2: Solve for x

$23+19=9x-3x$ → $42=6x$ → $x=7$.

Step3: Find JK

$JK=3(7)+23=21+23=44$.

Step4: Find PK

$PK$ is half of $JK$, so $PK=\frac{1}{2}JK=22$.

Problem 12

Step1: Set arcs equal

$DH=HE$, so $m\overset{\frown}{BG}=m\overset{\frown}{GC}$ → $9x-20=5x+28$.

Step2: Solve for x

$9x-5x=28+20$ → $4x=48$ → $x=12$.

Step3: Find $m\overset{\frown}{BG}$

$m\overset{\frown}{BG}=9(12)-20=108-20=88^\circ$.

Step4: Find $m\overset{\frown}{AB}$

Total circle is $360^\circ$, so $m\overset{\frown}{AB}=360^\circ - 2\times88^\circ=184^\circ$.

Problem 13

Step1: Find LN

Radius $LK=15$, $LN=9$, so $NK=\sqrt{LK^2-LN^2}=\sqrt{15^2-9^2}=\sqrt{225-81}=\sqrt{144}=12$.

Problem 14

Step1: Find $\angle KLN$

$\cos\angle KLN=\frac{LN}{LK}=\frac{9}{15}=0.6$, so $\angle KLN=53.13^\circ$, so $m\overset{\frown}{MK}=2\times53.13^\circ=106.26^\circ\approx106^\circ$.

Problem 15

Step1: Find JK

$JK=2\times NK=2\times12=24$.

Problem 16

Step1: Find $m\overset{\frown}{JPK}$

Total circle minus $m\overset{\frown}{MK}$, so $m\overset{\frown}{JPK}=360^\circ - 106^\circ=254^\circ$.

Answer:

Problem 9

$ED=26$
$CF=13$
$m\overset{\frown}{ED}=136^\circ$
$m\overset{\frown}{HD}=68^\circ$
$m\overset{\frown}{CE}=88^\circ$

Problem 10

$QU=8$
$QR=16$
$m\overset{\frown}{ST}=34^\circ$
$m\overset{\frown}{QR}=82^\circ$
$m\overset{\frown}{XT}=17^\circ$

Problem 11

$PK=22$

Problem 12

$m\overset{\frown}{AB}=184^\circ$

Problem 13

$NK=12$

Problem 14

$m\overset{\frown}{MK}\approx106^\circ$

Problem 15

$JK=24$

Problem 16

$m\overset{\frown}{JPK}=254^\circ$