Question

1. triangle lst is shown, where \\(\overline{mg}\\) is the midsegment. \\(sl = 2x + 6\\), \\(st = 3x + 10\\), \\(lt = 8x - 14\\), and \\(mg = 2x - 6\\). find the value of \\(x\\).

Explanation:

Step1: Apply Midsegment Theorem

By the triangle midsegment theorem, the midsegment $MG$ is half the length of the side $LT$ it is parallel to.
$$MG = \frac{1}{2}LT$$

Step2: Substitute given expressions

Replace $MG$ with $2x-6$ and $LT$ with $8x-14$.
$$2x - 6 = \frac{1}{2}(8x - 14)$$

Step3: Simplify right-hand side

Distribute $\frac{1}{2}$ across the parentheses.
$$2x - 6 = 4x - 7$$

Step4: Isolate $x$ terms

Subtract $2x$ from both sides.
$$-6 = 2x - 7$$

Step5: Solve for $x$

Add 7 to both sides, then divide by 2.
$$1 = 2x \implies x = \frac{1}{2}$$

Answer:

$\frac{1}{2}$