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The diagonals $A C$ and $B D$ and of a rectangle $A B C D$ of a rectangle intersect each other at $P$ intersect each other at . If $∠ A B D = 50 °,$ . If then the value of $∠ D P C$ then the value of from the following choices is:

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70°

90°

80°

100°

answer is C.

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Detailed Solution

Given that,

∠ A B D = 50 ° .

Construct a rhombus

A B C D

where the diagonals

A C = 24

cm, and

B D = 18

cm.

The diagonals

A C, B D

of a rectangle are equal, and they bisect each other.

A C = B D ⇒ 12 A C = 12 B D ⇒ A P = P B

Thus

Δ A P B

is an isosceles triangle,

∠ A B P = ∠ B A P = 50 °

.......(1)
[Angles opposite to equal sides are equal]
Calculate the value of

∠ A P B

.
The sum of angles of a triangle is

180 °

.
For the triangle APB,

∠ A B P + ∠ B A P + ∠ A P B = 180 ° ⇒ ∠ A P B + 50 ° + 50 ° = 180 ° ⇒ ∠ A P B = 80 ° ....... (2)

∠ A P B

and

∠ D P C

are opposite angles since they are formed by the intersection of

B D a n d A C

.
Calculate the value of

∠ D P C

.
From (2),

∠ A P B = ∠ D P C ⇒ ∠ D P C = 80 °

The value of

∠ D P C

.
Hence, option 3) is correct.

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