Question

1. if p is the circumcenter of $\triangle abc$, $ad = 3x - 11$, $db = 5x - 29$, and $pc = 18$, find $dp$.

$dp =$

1. if z is the incenter of $\triangle tuv$, find $mangle ztv$.

$mangle ztv =$
use the diagram below for questions 21-24 in which g is the centroid of $\triangle abc$,
$fc = 35$, $ag = 42$,
$bf = 57$ and $dg = 14$.

1. find $ac$.

$ac =$

1. find $bg$.

$bg =$

1. find $gc$.

$gc =$

1. find $ae$.

$ae =$

Explanation:

---

Question 19

Step1: Set AD=DB (circumcenter property)

$3x - 11 = 5x - 29$

Step2: Solve for x

$-2x = -18 \implies x=9$

Step3: Calculate DB length

$DB=5(9)-29=45-29=16$

Step4: Use PB=PC=18, right triangle PDB

$DP=\sqrt{PB^2 - DB^2}=\sqrt{18^2 - 16^2}=\sqrt{324-256}=\sqrt{68}=2\sqrt{17}$
---

Question 20

Step1: Set angle bisector equations equal

$5x+1=8x-14$

Step2: Solve for x

$-3x=-15 \implies x=5$

Step3: Find $\angle TVU$

$\angle TVU=2(8(5)-14)=2(26)=52^\circ$

Step4: Calculate $\angle UTV + \angle TUV$

$180^\circ - 21^\circ -52^\circ=107^\circ$

Step5: Find $\angle ZTV$ (half of $\angle UTV$)

$\angle ZTV=\frac{1}{2}(107^\circ)=53.5^\circ$
---

Question 21

Step1: AC=2FC (centroid, median property)

$AC=2\times35=70$
---

Question 22

Step1: BG=2DG (centroid ratio 2:1)

$BG=2\times14=28$
---

Question 23

Step1: FG=BF-BG, then GC=2FG

$FG=57-28=29 \implies GC=2\times29=58$
---

Question 24

Step1: AG=2GE, find GE

$GE=\frac{42}{2}=21$

Step2: AE=AG+GE

$AE=42+21=63$
---

Answer:

1. $DP = 2\sqrt{17}$
2. $m\angle ZTV = 53.5^\circ$
3. $AC = 70$
4. $BG = 28$
5. $GC = 58$
6. $AE = 63$