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State true or false.

Given parallelogram ABCD such that $∠ A = 60 °$ and point P is the midpoint of the side CD. Then DC = 2AD.

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True

False

answer is A.

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Detailed Solution

Given is a parallelogram ABCD such that

∠ A = 60 °

and point P is the midpoint of side CD.

∠ A = ∠ C = 60 °

, as opposite angles of parallelogram are equal.
Now, we know that the sum of all angle of quadrilateral is

360 °

∠ A + ∠ B + ∠ C + ∠ D = 360 ° ∠ B + ∠ D = 360 ° − 60 ° + 60 ° 2 ∠ B = 240 ° ∠ B = ∠ D = 120 °

As AB||DP and AP is a transversal we have,

∠ A P D = ∠ P A B ∠ A P D = 60 ° 2 ∠ A P D = ∠ P A B = 30 °

... (i)
Consider AB||PC and BP is a transversal,

∠ A B P = ∠ C P B ∠ A B P = ∠ B 2 = 120 ° 2 ∠ A B P = ∠ C P B = 60 °

... (ii)
Now as DPC is a straight line,

∠ A P D + ∠ A P B + ∠ C P B = 180 ° 30 ° + ∠ A P B + 60 ° = 180 ° ∠ A P B = 90 °

Consider equation (i),

∠ A P D = 30 ° ∠ D A P = 60 ° 2 = 30 ° ∠ A P D = ∠ D A P

... (iii)
Consider the

Δ A P D

∠ A P D = ∠ D A P D P = A D

… [Isosceles triangles]

D P = 12 D C

… [DP=PC where P is the midpoint of DC]
So, DC = 2AD.
So, the given statement is true.
Hence, the correct option is 2.

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