Question

ABCD is a parallelogram. DP and BQ are perpendiculars from D and B on AB and AD, respectively. Given AB = 16cm, BC = 10.8cm, DP = 9cm, find BQ.
2 points
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Answer 1

Step-by-step explanation:

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Answer 2

Answer:

and BQ are perpendiculars from D and B respectively on diagonal AC.

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Show that

(i)ΔADP≅ΔCBQ

(ii)∠ADP=∠CBQ

(iii)DP=BQ

Last updated date: 15th Feb 2023

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Answer

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Hint: ABCD is rectangle which means its opposite sides are equal. We can prove ΔADP≅ΔCBQ by using AAS criterion according to which if two angles and one side of a triangle are equal to the two angles and one side of another triangle then the triangles are congruent. Then by CPCT we can prove ∠ADP=∠CBQ and side DP=BQ .

Complete step-by-step answer:

Given ABCD is a rectangle then AB=CD and AD=BC

And DP and BQ are perpendiculars from D and B respectively on diagonal AC.

Then ∠P=∠Q=90∘ and two triangles ΔADP and ΔCBQ are formed.

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Now we have to prove-

(i)We have to prove that ΔADP and ΔCBQare congruent.

In ΔADP and ΔCBQ ,

∠APD=∠CQB=90∘ (Given)

∠DAP=∠BCQ (Because they are alternate angles)

Since opposite sides of rectangle are equal so we can write,

Side AB=side BC

Hence by Angle-Angle-Side congruence

ΔADP≅ΔCBQ Hence Proved

(ii) We have to prove that∠ADP=∠CBQ

Since we have already proved thatΔADP≅ΔCBQ

We know that corresponding parts of two congruent triangles are always equal.

Then by CPCT (Corresponding Parts of Congruent Triangles)

∠ADP=∠CBQ Hence Proved

(iii)We have to prove that side DP=side BQ

Since we have already proved that ΔADP≅ΔCBQ

We know that corresponding parts of two congruent triangles are always equal.

Then by CPCT (Corresponding Parts of Congruent Triangles)

Side DP=side BQ Hence Proved.

Note: AAS is one of the triangle congruence criterion which is used to prove that two triangles have equal sides and equal angles. The other congruence criterion are- SSS(side-side-side), ASA(angle-side-angle), SAS(side-angle-side) and HL(hypotenuse-leg).We cannot use ASA criterion because AD and BC are not include between the equal angles