Question

In the given figure, 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.

(i) Prove that point C is mid point of AQ.

(ii) Find the perimeter of quadrilateral BCQP.
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Answer 1

Given: ABCD is a  parallelogram. AB = 8 cm, AC = 10 cm, AD = 5 cm, AB is   produced to P, AB = BP.

To Find: (i) Prove that point C is midpoint of AQ.

             (ii) Find the perimeter of quadrilateral BCQP.

Step-by-step explanation:

(i) In ΔABC and ΔAPQ

      ∠B = ∠P  (corresponding angle are equal since BC║ PQ)

      ∠C = ∠Q (corresponding angle are equal since BC║ PQ)

       ∴ ΔABC ≅ ΔAPQ (By AA congruence)

   By CPCT

        $\frac{AB}{AP} = \frac{AC}{AQ}\\\frac{8}{16}=\frac{AC}{AQ} ( AB = 8 and AB = BP, AP = AB+BP = 8+8 = 16 )(given)\\AQ = 2\TIMES AC\\AC = \frac{1}{2} AQ$

∴ C is the midpoint of AQ.

(ii) Perimeter of quadrilateral BCQP = BC+CQ+QP+BP

   BC = AD = 5cm ( opposite sides of parallelogram are equal)

   CQ = AC = 10cm (C is the midpoint of AQ)

   AB = BP = 8cm (B is the midpoint of AP)

   By CPCT (ΔABC ≅ ΔAPQ)

   $\frac{AB}{BC}=\frac{AP}{PQ}\\\frac{8}{5}=\frac{16}{PQ}\\PQ = \frac{16\times 5}{8}\\PQ = 10cm$    

Perimeter = 8 + 10 + 10 + 5 = 23cm

Perimeter of quadrilateral BCQP = 23cm

   

Answer 2

Step-by-step explanation:

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