analytic geometry
module 6 test review
1. what is the length of $\overline{ab}$?
$\overline{ab} = \underline{\ 19\ }$
$8y - 5 = 11 - 14$
$3y + 5 = 14$
$3y - 5 - 5$
$3y = 9\ x = 3\ 8(3) - 5 = 19$
2. in $\triangle wxz$, what is the length of $\overline{zy}$?
$\overline{zy} = \underline{\ 4.8\ }$
3. in the figure, $g$ is the circumcenter of $\triangle ace$. what is the length of $\overline{ce}$ to the nearest tenth?
$\overline{ce} = \underline{\ 13.4\ }$
$x^2 + 5.2^2 = 8.5^2$
$x^2 = 8.5^2 - 5.2^2$
$x = 6.7$
$6.7 + 6.7 = 13.4$
4. what is the length of $\overline{rs}$?
$\overline{rs} = \underline{\ 23\ }$
$7x + 2 = 3x + 14$
$4x + 2 = 14$
$4x + 2 - 2 = 14 - 2$
$4x = 12$
$x = 3$
$7(3) + 2 = 23$
5. true or false: using the converse of the angle bisector theorem, the $m\angle efh = 27^\circ$.
(marked as true)
6. in the figure, $g$ is the incenter of $\triangle ace$. what is the length of $\overline{gb}$?
$\overline{gb} = \underline{\ 8\ }$
$x^2 + 15^2 = 17^2$
$x^2 = \sqrt{17^2 - 15^2}$
$x = 8$
7. in $\triangle abc$, $g$ is the centroid and $\overline{dc} = 36$. find $\overline{dg}$ and $\overline{gc}$.
$\overline{dg} = \underline{\ 12\ }$
$\overline{gc} = \underline{\ 24\ }$
$13 = 12$
8. in $\triangle rst$, $d$ is the centroid and $\overline{db} = 18$. find $\overline{rd}$.
$\overline{rd} = \underline{\ 36\ }$

Question

Accepted Answer

### Problem 1: Length of $\overline{AB}$
# Explanation:
## Step 1: Set up the equation
Since $AC$ is the perpendicular bisector of $BD$, $AB = AD$. So $8y - 5=11y - 14$.
## Step 2: Solve for $y$
Rearrange the equation: $11y - 8y=14 - 5$ → $3y = 9$ → $y = 3$.
## Step 3: Calculate $AB$
Substitute $y = 3$ into $8y - 5$: $8(3)-5=24 - 5 = 19$.
# Answer: $19$

### Problem 2: Length of $\overline{ZY}$
# Explanation:
## Step 1: Use the property of isosceles triangle
In $\triangle WXZ$, $WZ = WX = 9$, and $WY$ is the perpendicular bisector of $XZ$, so $ZY = XY$. Given $XY = 4.8$, so $ZY = 4.8$ (Note: The original handwritten "49" might be a typo, likely "4.8" as per the diagram).
# Answer: $4.8$

### Problem 3: Length of $\overline{CE}$
# Explanation:
## Step 1: Find $CD$
Since $G$ is the circumcenter, $GD$ is perpendicular to $CE$ and bisects it. First, use the Pythagorean theorem in $\triangle CGD$: Let $CD=x$, $CG = 8.5$, $GD = 5.2$. So $x^{2}+5.2^{2}=8.5^{2}$.
## Step 2: Solve for $x$
$x^{2}=8.5^{2}-5.2^{2}=72.25 - 27.04 = 45.21$ → $x=\sqrt{45.21}\approx6.7$.
## Step 3: Calculate $CE$
Since $CD = DE$, $CE = 2x=2\times6.7 = 13.4$.
# Answer: $13.4$

### Problem 4: Length of $\overline{RS}$
# Explanation:
## Step 1: Set up the equation
Since $QS$ is the angle bisector and $QR\perp RS$, $QT\perp ST$, so $RS = TS$? Wait, actually, since $7x + 2$ and $3x + 14$ are equal (because $S$ is on the angle bisector, so distances from $S$ to the sides are equal? Wait, no, the segments $7x + 2$ and $3x + 14$ are equal. So $7x+2 = 3x + 14$.
## Step 2: Solve for $x$
$7x-3x=14 - 2$ → $4x = 12$ → $x = 3$.
## Step 3: Calculate $RS$
Substitute $x = 3$ into $7x + 2$: $7(3)+2=21 + 2 = 23$.
# Answer: $23$

### Problem 5: True or False
# Explanation:
## Step 1: Apply the converse of angle bisector theorem
The converse of the angle bisector theorem states that if a point is equidistant from the sides of an angle, it lies on the angle bisector. Here, $FH$ is equidistant from $FE$ and $FG$ (perpendicular distances), so $FH$ bisects $\angle EFG$. If $\angle HFG = 27^{\circ}$, then $\angle EFH = 27^{\circ}$, so the statement is True.
# Answer: True

### Problem 6: Length of $\overline{GB}$
# Explanation:
## Step 1: Use the property of incenter
$G$ is the incenter, so $GB = GD = GF$ (distances from incenter to the sides). In $\triangle GDC$, use Pythagorean theorem: Let $GB=x$, $GC = 17$, $DC = 15$. So $x^{2}+15^{2}=17^{2}$.
## Step 2: Solve for $x$
$x^{2}=17^{2}-15^{2}=289 - 225 = 64$ → $x = 8$.
# Answer: $8$

### Problem 7: Length of $\overline{DG}$ and $\overline{GC}$
# Explanation:
## Step 1: Property of centroid
The centroid $G$ divides the median $DC$ in the ratio $1:2$, so $DG:GC = 1:2$ and $DG+GC = DC = 36$.
## Step 2: Solve for $DG$ and $GC$
Let $DG = x$, $GC = 2x$. Then $x + 2x=36$ → $3x = 36$ → $x = 12$. So $DG = 12$, $GC = 24$.
# Answer: $DG = 12$, $GC = 24$

### Problem 8: Length of $\overline{RD}$
# Explanation:
## Step 1: Property of centroid
The centroid $D$ divides the median $RB$ in the ratio $2:1$, so $RD:DB = 2:1$. Given $DB = 18$, then $RD = 2\times18 = 36$.
# Answer: $36$

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QUESTION IMAGE

Question

Explanation:

Problem 1: Length of \(\overline{AB}\)

Step 1: Set up the equation

Step 2: Solve for \(y\)

Step 3: Calculate \(AB\)

Step 1: Use the property of isosceles triangle

Step 1: Find \(CD\)

Step 2: Solve for \(x\)

Step 3: Calculate \(CE\)

Answer:

Problem 2: Length of \(\overline{ZY}\)