Question

b. incenter
c. centroid
d. orthocenter

1. classify point c.

a. circumcenter
b. incenter
c. centroid
d. orthocenter

c. centroid
d. orthocent

1. classify point c

a. circumcenter
b. incenter
c. centroid
d. orthocenter

use the diagram below for questions 13-15 in which h is the circumcenter of $\triangle bcd$, $bc = 18$ and $hd = 14$.

1. find $be$.
2. find $ch$.
3. find $eh$.

use the diagram below for questions 16-18 in which p is the incenter of $\triangle jkl$, $pn = 21$ and $ml = 27$.

1. find $po$.
2. find $lo$.
3. find $pl$.

Explanation:

Step1: Solve for BE (Q13)

H is circumcenter, so BE is half BC.
$BE = \frac{BC}{2} = \frac{18}{2} = 9$

Step2: Solve for CH (Q14)

Circumcenter is equidistant from vertices, so $CH = HD = 14$

Step3: Solve for EH (Q15)

Use Pythagorean theorem in $\triangle BEH$: $EH = \sqrt{BH^2 - BE^2}$. $BH=HD=14$, so
$EH = \sqrt{14^2 - 9^2} = \sqrt{196 - 81} = \sqrt{115}$

Step4: Solve for PO (Q16)

P is incenter, so $PO = PN = 21$

Step5: Solve for LO (Q17)

Incenter creates congruent tangent segments, so $LO = ML = 27$

Step6: Solve for PL (Q18)

Use Pythagorean theorem in $\triangle PLO$: $PL = \sqrt{LO^2 + PO^2}$
$PL = \sqrt{27^2 + 21^2} = \sqrt{729 + 441} = \sqrt{1170} = 3\sqrt{130}$

Answer:

1. $9$
2. $14$
3. $\sqrt{115}$
4. $21$
5. $27$
6. $3\sqrt{130}$