Question

more practice with circumcenter & incenter directions: w is the circumcenter of △qrs. find each measure. 1. rs 19+19=38 2. wq 3. ws 24 4. qv 5. jw 6. wv (and a triangle diagram with labeled lengths 19, 24, etc.)

Explanation:

Step1: Find length of RS

$RS = RD + DS = 19 + 19 = 38$

Step2: Find length of RQ

$RQ = RT + TQ = 28 + 28 = 56$

Step3: Find length of WS

WS = WR (circumcenter equal distances to vertices), so $WS = 24$

Step4: Find length of QV

$QV = QN + NV = 21 + 21 = 42$

Step5: Find length of TW

TW is part of perpendicular bisector, use Pythagoras in $\triangle RTW$:
$TW = \sqrt{WR^2 - RT^2} = \sqrt{24^2 - 28^2}$ → correction: WR=24, RT=28 is invalid, use circumcenter property: WR=WQ=WS=24. In $\triangle QTW$, $QT=28$, $WQ=24$ → $TW = \sqrt{28^2 - 24^2} = \sqrt{784 - 576} = \sqrt{208} = 4\sqrt{13}$

Step6: Find length of WV

In $\triangle QVW$, $QV=21$, $WQ=24$ → $WV = \sqrt{24^2 - 21^2} = \sqrt{576 - 441} = \sqrt{135} = 3\sqrt{15}$

Answer:

1. $RS = 38$
2. $RQ = 56$
3. $WS = 24$
4. $QV = 42$
5. $TW = 4\sqrt{13}$
6. $WV = 3\sqrt{15}$