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The problem you’ve uploaded involves finding several unknowns related to a triangle. The key piece of information is that \( \overline{PS} \) is the median of triangle \( \triangle PQR \). This means that \( \overline{PS} \) divides the triangle into two equal areas, and also \( \overline{QS} = \overline{SR} \), i.e., \( \overline{QS} \) and \( \overline{SR} \) are of equal length.

The angles and segment lengths are expressed as algebraic expressions:
- \( \angle QPS = 15x + 42^\circ \)
- \( \overline{QS} = 10x - 7 \)
- \( \overline{SR} = 5x + 3 \)

We need to find the value of \( x \), the lengths of \( \overline{SR} \) and \( \overline{QR} \), as well as the measure of angle \( \angle PSR \).

Let’s proceed step-by-step to solve for each variable:

### Step 1: Set up the equation for \( x \) using the fact that \( \overline{QS} = \overline{SR} \)
Since \( \overline{PS} \) is a median, we know that:
\[
\overline{QS} = \overline{SR}
\]
Thus, we can equate the expressions for \( \overline{QS} \) and \( \overline{SR} \):
\[
10x - 7 = 5x + 3
\]
Solving for \( x \):

### Step 2: Calculate the lengths of the sides and the measure of \( \angle PSR \)
Once we have the value of \( x \), we can substitute it into the expressions for \( \overline{SR} \) and \( \overline{QR} \).

Let me work through the calculations now. sympy import symbols, Eq, solve

# Define the variable
x = symbols('x')

# Set up the equation for 10x - 7 = 5x + 3
equation = Eq(10*x - 7, 5*x + 3)

# Solve for x
x_value = solve(equation, x)

x_valueThe value of \( x \) is 2. 

### Step 3: Calculate the lengths of \( \overline{SR} \) and \( \overline{QR} \)

Now that we know \( x = 2 \), we can substitute this value into the expressions for \( \overline{SR} \) and \( \overline{QR} \):

- \( \overline{SR} = 5x + 3 \)
- \( \overline{QR} = 10x - 7 \)

Let me compute those values. Calculate the lengths of SR and QR
SR = 5*x_value[0] + 3
QR = 10*x_value[0] - 7

SR, QRThe lengths of \( \overline{SR} \) and \( \overline{QR} \) are both 13 units.

### Step 4: Find the measure of \( \angle PSR \)

We know that \( \angle QPS = 15x + 42^\circ \), and now that \( x = 2 \), we can calculate \( \angle QPS \):

\[
\angle QPS = 15(2) + 42^\circ = 30^\circ + 42^\circ = 72^\circ
\]

Since \( \overline{PS} \) is the median, the angles \( \angle QPS \) and \( \angle PSR \) are equal (as they form a straight line at point \( S \)).

Thus, the measure of \( \angle PSR \) is \( 72^\circ \).

### Final answers:
- \( x = 2 \)
- \( \overline{SR} = 13 \)
- \( \overline{QR} = 13 \)
- \( m\angle PSR = 72^\circ \)

Would you like more details or help with other questions? Here are some related questions:
1. What is the formula for the length of a median in a triangle?
2. How does the median divide the area of a triangle?
3. How do we prove that two angles are equal when they share a common vertex and a side?
4. What is the relationship between the sides of an isosceles triangle?
5. How do you calculate the angle between two sides of a triangle?

Tip: Remember that when working with geometric properties, recognizing symmetries (like medians and equal sides) can simplify calculations significantly.

PS is a median of ΔPQR. Find the following: x, SR, QR, and m∠PSR, given the diagram and expressions for QS, SR, and angle QPS.

Explore how to solve a triangle median problem with algebra and geometry. Learn to calculate x, the lengths of segments SR and QR, and the angle PSR using properties of medians and angle relationships. Suitable for students in Grades 8-10.

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