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If \( SR \) bisects \( PT \), \( PQ = 12x + 7 \), \( QT = 8x + 27 \), \( SR = 6x + 30 \), and \( QR = 23 \), find \( SQ \).

Answer :

The length of SQ is 76.

To solve the problem, we are given several pieces of information about a line segment bisected by another line segment.

PQ = 12x + 7

QT = 8x + 27

SR = 6x + 30

QR = 23

Since SR bisects PT, we can establish the following relationship:

PT = PQ + QT

Since SR bisects PT, SR = (1/2)PT

Now, we will first find the lengths of PQ and QT in terms of x.

Step 1: Set Up the Equation

From the information, we have:

[tex]PT = PQ + QT = (12x + 7) + (8x + 27)[/tex]
[tex]PT = 20x + 34[/tex]

Since SR bisects PT, we can say:
[tex]SR = \frac{1}{2}(PT) = \frac{1}{2}(20x + 34) = 10x + 17[/tex]

Step 2: Equate SR and the Given SR

Now we can set the expression for SR equal to the given value:
[tex]10x + 17 = 6x + 30[/tex]

Step 3: Solve for x

Now we solve for x:
[tex]10x - 6x = 30 - 17[/tex]
[tex]4x = 13[/tex]
[tex]x = \frac{13}{4}[/tex]

Step 4: Calculate Line Segment Lengths

We can now substitute x back into the equations for PQ and QT:

PQ:
[tex]PQ = 12x + 7 = 12 \left(\frac{13}{4}\right) + 7 = 39 + 7 = 46[/tex]

QT:
[tex]QT = 8x + 27 = 8 \left(\frac{13}{4}\right) + 27 = 26 + 27 = 53[/tex]

PT:
[tex]PT = PQ + QT = 46 + 53 = 99[/tex]

Step 5: Find SQ

Since QR = 23, and we want to find SQ, we also know that:
[tex]SQ = QS + QR[/tex]
Since QR is one of the segments that makes up the whole PT:
[tex]Q R = 23[/tex]
Thus:
[tex]SQ = PT - QR = 99 - 23 = 76[/tex]

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