Question

In trapezium $ABCD$, side $AB \parallel PQ \parallel DC$. If $AP = 3$, $PD = 12$, and $QC = 14$, find $BQ$. 

Hint:

In trapeziums with parallel sides, the intercept theorem (or basic proportionality theorem) can be used to find missing segment lengths.

Answer 1

Step 1: Understanding the given figure.
In trapezium $ABCD$, the sides $AB$, $PQ$, and $DC$ are parallel. The lines $AD$ and $BC$ are transversals. Hence, the ratios of the corresponding segments on these transversals are equal.
\[ \frac{AP}{PD} = \frac{BQ}{QC} \]
Step 2: Substitute the given values.
\[ \frac{3}{12} = \frac{BQ}{14} \]
Step 3: Simplify the equation.
\[ \frac{1}{4} = \frac{BQ}{14} \]
\[ BQ = \frac{14}{4} = 3.5 \]
Step 4: Conclusion.
Hence, $BQ = 3.5$ cm.
Correct Answer: $BQ = 3.5$ cm