Question

$overline{jk} parallel overline{gi}$. find $hk$.
$hk = square$

Explanation:

Step1: Apply Triangle Proportionality Theorem

Since $\overline{JK} \parallel \overline{GI}$, the segments created on the sides of the triangle are proportional:
$$\frac{HK}{KJ} = \frac{HI}{IJ}$$
Let $HK = x$. Then $KJ = x + 18$, $HI = 70$, $IJ = 28$.

Step2: Substitute values into proportion

$$\frac{x}{x + 18} = \frac{70}{28}$$
Simplify $\frac{70}{28}$ to $\frac{5}{2}$:
$$\frac{x}{x + 18} = \frac{5}{2}$$

Step3: Cross-multiply to solve for x

$$2x = 5(x + 18)$$
Expand the right side:
$$2x = 5x + 90$$
Rearrange terms to isolate x:
$$2x - 5x = 90$$
$$-3x = 90$$
$$x = \frac{90}{-3} = -30$$
Since length cannot be negative, reverse the proportion (we matched the wrong corresponding segments):
Correct proportion: $\frac{HK}{HG} = \frac{IJ}{JI}$
Let $HK = x$, so $HG = x - 18$, $JI = 28 + 70 = 98$
$$\frac{x}{x - 18} = \frac{98}{28}$$
Simplify $\frac{98}{28} = \frac{7}{2}$

Step4: Cross-multiply and solve

$$2x = 7(x - 18)$$
$$2x = 7x - 126$$
$$2x - 7x = -126$$
$$-5x = -126$$
$$x = \frac{126}{5} = 25.2$$
Wait, correct proportionality: When $\overline{GI} \parallel \overline{JK}$, $\triangle HGI \sim \triangle HKJ$ by AA similarity.
Thus $\frac{HG}{HK} = \frac{HI}{HJ}$
Let $HK = x$, $HG = x - 18$, $HJ = 70 + 28 = 98$, $HI = 70$
$$\frac{x - 18}{x} = \frac{70}{98}$$
Simplify $\frac{70}{98} = \frac{5}{7}$

Step5: Correct cross-multiplication

$$7(x - 18) = 5x$$
$$7x - 126 = 5x$$
$$7x - 5x = 126$$
$$2x = 126$$
$$x = 63$$

Answer:

$63$