3. find x.

4. find mn

5. find cd.

6. find jl.

7. if $\overline{qt}$ is the perpendicular bisector of $\overline{pr}$, find each measure.
$x = \underline{quadquad}$
$y = \underline{quadquad}$
$pq = \underline{quadquad}$
$qr = \underline{quadquad}$
$ps = \underline{quadquad}$
$sr = \underline{quadquad}$
$pt = \underline{quadquad}$
$pr = \underline{quadquad}$

Question

3. find x.

4. find mn

5. find cd.

6. find jl.

7. if $\overline{qt}$ is the perpendicular bisector of $\overline{pr}$, find each measure.
$x = \underline{quadquad}$
$y = \underline{quadquad}$
$pq = \underline{quadquad}$
$qr = \underline{quadquad}$
$ps = \underline{quadquad}$
$sr = \underline{quadquad}$
$pt = \underline{quadquad}$
$pr = \underline{quadquad}$

Accepted Answer

### Problem 3: Find $ x $
# Explanation:
## Step 1: Identify Equal Sides
The triangle has two equal sides (marked with ticks), so the two given expressions for the sides are equal: $ 8x - 37 = 5x + 17 $.

## Step 2: Solve for $ x $
Subtract $ 5x $ from both sides: $ 8x - 5x - 37 = 17 $
Simplify: $ 3x - 37 = 17 $
Add 37 to both sides: $ 3x = 17 + 37 $
Calculate: $ 3x = 54 $
Divide by 3: $ x = \frac{54}{3} = 18 $

# Answer:
$ x = 18 $

### Problem 4: Find $ MN $
# Explanation:
## Step 1: Identify Equal Sides
In the isosceles triangle (since $ MP = PO $ and $ PN $ is perpendicular), $ MN = ON $. So $ 4x + 19 = 6x - 7 $.

## Step 2: Solve for $ x $
Subtract $ 4x $ from both sides: $ 19 = 2x - 7 $
Add 7 to both sides: $ 26 = 2x $
Divide by 2: $ x = 13 $

## Step 3: Find $ MN $
Substitute $ x = 13 $ into $ 4x + 19 $: $ 4(13) + 19 = 52 + 19 = 71 $

# Answer:
$ MN = 71 $

### Problem 5: Find $ CD $
# Explanation:
## Step 1: Identify Equal Segments
Since $ AB = BC $ (ticks), $ AD = CD $. So $ 9x - 15 = 7x - 1 $.

## Step 2: Solve for $ x $
Subtract $ 7x $ from both sides: $ 2x - 15 = -1 $
Add 15 to both sides: $ 2x = 14 $
Divide by 2: $ x = 7 $

## Step 3: Find $ CD $
Substitute $ x = 7 $ into $ 7x - 1 $: $ 7(7) - 1 = 49 - 1 = 48 $

# Answer:
$ CD = 48 $

### Problem 6: Find $ JL $
# Explanation:
## Step 1: Identify Equal Segments
The triangle has two equal sides (ticks), so $ JM = ML $. Thus, $ 3x + 4 = 5x - 16 $.

## Step 2: Solve for $ x $
Subtract $ 3x $ from both sides: $ 4 = 2x - 16 $
Add 16 to both sides: $ 20 = 2x $
Divide by 2: $ x = 10 $

## Step 3: Find $ JL $
$ JL = JM + ML = (3x + 4) + (5x - 16) $. Substitute $ x = 10 $:
$ (3(10) + 4) + (5(10) - 16) = (34) + (34) = 68 $

# Answer:
$ JL = 68 $

### Problem 7: If $ \overline{QT} $ is the perpendicular bisector of $ \overline{PR} $
# Explanation:
#### Step 1: Find $ x $ (from $ PS = SR $)
$ PS = 4x + 4 $ and $ SR = 7x - 17 $. Set equal: $ 4x + 4 = 7x - 17 $
Subtract $ 4x $: $ 4 = 3x - 17 $
Add 17: $ 21 = 3x $
$ x = 7 $

#### Step 2: Find $ y $ (from $ PQ = QR $)
$ PQ = 5y - 31 $ and $ QR = 2y + 5 $. Set equal: $ 5y - 31 = 2y + 5 $
Subtract $ 2y $: $ 3y - 31 = 5 $
Add 31: $ 3y = 36 $
$ y = 12 $

#### Step 3: Find $ PQ $
Substitute $ y = 12 $ into $ 5y - 31 $: $ 5(12) - 31 = 60 - 31 = 29 $

#### Step 4: Find $ QR $
Substitute $ y = 12 $ into $ 2y + 5 $: $ 2(12) + 5 = 24 + 5 = 29 $ (matches $ PQ $, good)

#### Step 5: Find $ PS $
Substitute $ x = 7 $ into $ 4x + 4 $: $ 4(7) + 4 = 28 + 4 = 32 $

#### Step 6: Find $ SR $
Substitute $ x = 7 $ into $ 7x - 17 $: $ 7(7) - 17 = 49 - 17 = 32 $ (matches $ PS $, good)

#### Step 7: Find $ PT $ (and $ TR $, but $ PT = TR $ as bisector)
First, find $ 6x - 2y $: $ 6(7) - 2(12) = 42 - 24 = 18 $. Since $ QT $ is perpendicular bisector, $ PT = TR $, but we can also use $ PT $ from $ 6x - 2y = 18 $. Wait, actually, $ PT $ is part of $ PR $, but we already found $ PS = SR = 32 $, so $ PR = 64 $, so $ PT = \frac{PR}{2} = 32 $? Wait, no, $ PT $ is from the expression $ 6x - 2y = 18 $, which is $ PT $. Let's confirm:

$ PT = 6x - 2y = 6(7) - 2(12) = 42 - 24 = 18 $

#### Step 8: Find $ PR $
$ PR = PS + SR = 32 + 32 = 64 $ (or $ PR = 2 \times PT = 2 \times 18 = 36 $? Wait, no, mistake here. Wait, $ PS = 4x + 4 = 32 $, $ SR = 7x - 17 = 32 $, so $ PR = PS + SR = 64 $. But $ PT = 6x - 2y = 18 $, so $ TR = PT = 18 $ (since $ QT $ is bisector), so $ PR = PT + TR = 36 $. Contradiction? Wait, no, I messed up. $ PS $ and $ SR $ are sides of the quadrilateral, not $ PR $. $ PR $ is the segment bisected by $ QT $, so $ PT = TR $. And $ PS = SR $ (ticks on $ PS $ and $ SR $), so $ 4x + 4 = 7x - 17 $ gives $ x = 7 $, so $ PS = SR = 32 $. Then $ PQ = QR $ (since $ QT $ is perpendicular bisector, so $ \triangle PQR $ is isoceles), so $ 5y - 31 = 2y + 5 $ gives $ y = 12 $, so $ PQ = QR = 29 $. Then $ PT = 6x - 2y = 6(7) - 2(12) = 42 - 24 = 18 $, so $ TR = PT = 18 $, so $ PR = PT + TR = 36 $. Wait, but $ PS = 32 $, $ SR = 32 $, so $ PS = SR = 32 $, which is a different segment. So correct:

- $ x = 7 $
- $ y = 12 $
- $ PQ = 5y - 31 = 29 $
- $ QR = 2y + 5 = 29 $
- $ PS = 4x + 4 = 32 $
- $ SR = 7x - 17 = 32 $
- $ PT = 6x - 2y = 18 $
- $ PR = PT + TR = 18 + 18 = 36 $

# Answer:
- $ x = 7 $
- $ y = 12 $
- $ PQ = 29 $
- $ QR = 29 $
- $ PS = 32 $
- $ SR = 32 $
- $ PT = 18 $
- $ PR = 36 $

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Problem 3: Find \( x \)

Step 1: Identify Equal Sides

Step 2: Solve for \( x \)

Step 1: Identify Equal Sides

Step 2: Solve for \( x \)

Step 3: Find \( MN \)

Step 1: Identify Equal Segments

Step 2: Solve for \( x \)

Step 3: Find \( CD \)

Answer:

Problem 4: Find \( MN \)