In the given figure ABCD is parallelogram. AB is produced to P, such that AB= BP and PQ is parallel to BC to meet AC produced at Q. Given AB =8 cm, AD=5 cm,AC=10. 1. prove that point C is mid point of AQ. 2. find the perimeter of quadrilateral BCQP.
Answers
Answer:
ABCD is a parallelogram. AB is produced to P, such that AB = BP and PQ is drawn parallel to BC to meet AC produced at Q. Given AB = 8 cm, AD = 5 cm, AC = 10 cm.
→ BC ║ QP →→Given
In Δ ABC and ΔAPQ
∠ABC=∠APQ→→[BC ║ QP , BP is a transversal, so corresponding angles are equal]
∠BAC=∠PAQ→→Reflex angle
Δ ABC ~ ΔAPQ→→(AA similarity criterion]
When triangles are similar, their corresponding angles are equal.
\begin{gathered}\frac{AB}{AP}=\frac{AC}{AQ}=\frac{BC}{PQ}\\\\ \frac{8}{16}=\frac{10}{10+QC}=\frac{5}{PQ}\\\\ QC=20-10=10\\\\ PQ=5 \times 2\\\\ PQ=10\end{gathered}
AP
AB
=
AQ
AC
=
PQ
BC
16
8
=
10+QC
10
=
PQ
5
QC=20−10=10
PQ=5×2
PQ=10
AC=CQ=10 cm, shows that point C is mid point of AQ.
(b) Perimeter of quadrilateral BCQP=BC +CQ+QP+BP
= 5 + 10 +10+8
= 33 cm