Question

find jk.
jk = □
ig = 17, ij = 15, gk = 34

Explanation:

Step1: Identify similar triangles

The figure shows $\triangle IHK \sim \triangle JGK$ (by the Basic Proportionality Theorem, since $IH \parallel JG$).

Step2: Set up proportion of sides

Corresponding sides are proportional:
$$\frac{JK}{IK} = \frac{GK}{HK}$$
First calculate $HK = HG + GK = 17 + 34 = 51$, and $IK = IJ + JK = 15 + JK$.

Step3: Substitute values into proportion

$$\frac{JK}{15 + JK} = \frac{34}{51}$$
Simplify $\frac{34}{51} = \frac{2}{3}$, so:
$$\frac{JK}{15 + JK} = \frac{2}{3}$$

Step4: Solve for $JK$

Cross-multiply to eliminate fractions:
$$3 \cdot JK = 2 \cdot (15 + JK)$$
$$3JK = 30 + 2JK$$
Subtract $2JK$ from both sides:
$$3JK - 2JK = 30$$
$$JK = 30$$

Answer:

$30$