Final answer:
To find x, set up the equation 3x-5 = x+13 and solve for x. To find PS, find QS first by doubling RS, then add PQ and QS.
Step-by-step explanation:
To find the value of x, we can set up an equation using the fact that Q is the midpoint of PR. Since Q is the midpoint, we know that PQ is equal to QR. So we can set up the equation: 3x-5 = x+13. Solving this equation, we can find the value of x by subtracting x from both sides and adding 5 to both sides: 3x-x = 13+5. This simplifies to 2x = 18, and dividing both sides by 2, we get x = 9.
To find the length of PS, we need to find the length of QS first. Since R is the midpoint of QS, we know that QS is equal to 2 times the length of RS. Since QR is given as x+13, RS is half of that, so it is equal to (x+13)/2. Therefore, QS is equal to 2 times RS, which is 2 times (x+13)/2. Simplifying, we get QS = x+13.
Now, to find the length of PS, we need to add PQ and QS. PQ is given as 3x-5, and QS is x+13. Adding these, we get PS = PQ + QS = (3x-5) + (x+13). Simplifying, we get PS = 4x + 8.