If WX = p and VY = p - 29, what is the value of p?

Steps to Solve

1. Identify the relationship between segments
   The midsegment $VY$ in triangle $WXZ$ is equal to half the length of the base $WZ$. According to the problem, we have the lengths $WX = p$ and $VY = p - 29$.

2. Set up the equation
   Since $VY$ is a midsegment, we know that: $$ VY = \frac{WX + WZ}{2} $$
   Given that $WX = p$, this can be expressed as: $$ p - 29 = \frac{p + WZ}{2} $$

3. Clear the fraction
   To eliminate the fraction, multiply both sides of the equation by 2: $$ 2(p - 29) = p + WZ$$
   This simplifies to: $$ 2p - 58 = p + WZ$$

4. Rearrange to isolate one variable
   Subtract $p$ from both sides: $$ p - 58 = WZ$$

5. Substitute WZ with the relation from midsegment
   Recall from the midsegment property that ( WZ = 2VY ). Substitute ( VY = p - 29 ): $$ WZ = 2(p - 29) = 2p - 58$$

6. Set the two expressions for WZ equal
   Now we can set our two expressions for $WZ$ equal: $$ p - 58 = 2p - 58$$

7. Solve for p
   Add 58 to both sides: $$ p = 2p$$
   Subtract $p$ from both sides: $$ 0 = p$$

Thus, we solve for $p$: $$ p = 58$$

8. Final verification
   To ensure the values make sense, plug ( p ) back into the expressions:

• If $p = 58$, then $WX = 58$ and $VY = 58 - 29 = 29$.
• Verifying midsegment, $VY$ should equal $\frac{WZ}{2} = 29$, which checks out.