Question

gk || hj. find y.
g, k, h, j, i are points. gh = 4, hi = 3, kj = 5y - 2, ij = 9y - 12. gk is parallel to hj. find y.
y =

Explanation:

Step1: Apply Basic Proportionality Theorem

Since \( GK \parallel HJ \), by the Basic Proportionality Theorem (Thales' theorem), the ratio of the segments on one side is equal to the ratio of the segments on the other side. So, \(\frac{GH}{HI}=\frac{KJ}{JI}\).
Substituting the given values: \(\frac{4}{3}=\frac{5y - 2}{9y - 12}\).

Step2: Cross - Multiply to Solve for \( y \)

Cross - multiplying gives \( 4(9y - 12)=3(5y - 2) \).
Expanding both sides: \( 36y-48 = 15y-6 \).
Subtract \( 15y \) from both sides: \( 36y - 15y-48=15y - 15y-6 \), which simplifies to \( 21y-48=-6 \).
Add 48 to both sides: \( 21y-48 + 48=-6 + 48 \), so \( 21y = 42 \).
Divide both sides by 21: \( y=\frac{42}{21}=2 \).

Answer:

\( y = 2 \)