Question: 13. Given H is the midpoint of JK, solve for x and find JH, HK and JK.
J —— H —— K
left segment is labeled 4x - 15 and right segment is labeled 2x + 3; position_hint=top-center
1. **State the problem:**
We are given that point $H$ is the midpoint of segment $JK$. The segment $JK$ is divided into two parts: $JH$ and $HK$.
2. **Understand the midpoint property:**
Since $H$ is the midpoint, the lengths of $JH$ and $HK$ are equal.
3. **Set up the equation:**
Given:
$$JH = 4x - 15$$
$$HK = 2x + 3$$
Since $H$ is the midpoint, we have:
$$4x - 15 = 2x + 3$$
4. **Solve for $x$:**
Subtract $2x$ from both sides:
$$4x - 2x - 15 = 2x - 2x + 3$$
$$2x - 15 = 3$$
Add $15$ to both sides:
$$2x - 15 + 15 = 3 + 15$$
$$2x = 18$$
Divide both sides by $2$:
$$\cancel{2}x = \frac{18}{\cancel{2}}$$
$$x = 9$$
5. **Find lengths $JH$ and $HK$:**
Substitute $x=9$ into $JH$:
$$JH = 4(9) - 15 = 36 - 15 = 21$$
Substitute $x=9$ into $HK$:
$$HK = 2(9) + 3 = 18 + 3 = 21$$
6. **Find length $JK$:**
Since $JK = JH + HK$:
$$JK = 21 + 21 = 42$$
**Final answers:**
$$x = 9$$
$$JH = 21$$
$$HK = 21$$
$$JK = 42$$