ABCD is a trapezium in which AB||DC and P and Q arepoints on AD and BC respectively such that PQ||DC .If PD = 12 cm, BQ = 42 cm and QC = 18 cm, find AD.
Question
ABCD is a trapezium in which AB||DC and P and Q arepoints on AD and BC respectively such that PQ||DC .If PD = 12 cm, BQ = 42 cm and QC = 18 cm, find AD.
Solution
Given that ABCD is a trapezium with AB||DC and PQ||DC. Also, PD = 12 cm, BQ = 42 cm, and QC = 18 cm.
Since PQ is parallel to DC and DC is parallel to AB, by the transitive property, PQ is also parallel to AB.
This means that triangle PDQ and triangle BQC are similar (by the AA postulate, as they have two pairs of corresponding angles equal).
The ratio of corresponding sides in similar triangles is equal. Therefore, we can write:
PD/BQ = DQ/QC
Substituting the given values:
12/42 = DQ/18
Solving for DQ, we get:
DQ = (12 * 18) / 42 = 5.14 cm (approximately)
Now, AD = PD + DQ = 12 cm + 5.14 cm = 17.14 cm
So, AD = 17.14 cm.
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