Question

ABCD is a parallelogram. P and Q are points
on the diagonal AC such that DP and BQ are
perpendiculars to AC.
Prove that DP=BQ

Answer 1

$\sf\small\underline{Given:-}$

ABCD is parallelogram. P and Q are points on the diagonal AC such that DP and BQ are Perpendicular on AC.

$\sf\small\underline{Prove\:That:-}$

$\rm{\leadsto DP=BQ}$

$\sf\small\underline{Solution:-}$

To prove DP = BQ , At first draw perpendicular DP and BQ on the diagonal of parallelogram on AC , such that DP _|_ AC and BQ _|_ AC.

$\sf\small\underline{Calculation\:begins:-}$

$\rm{\implies As\:per\:the\: above\: figure:-}$

We will take two traingle to prove congruency. ∆ ADP and ∆ BQC . In parallelogram ABCD with diagonal AC on which two perpendicular DP and BQ lies such that two triangle formed ∆ ADP and ∆ BQC. Let's prove congruency now.

$\rm{\leadsto In\:\triangle\:ADP\:and\:\triangle\:BQC}$

$\rm{\leadsto \angle\:APD=\angle\:BQC\:(each\:90\degree)}$

$\rm{\leadsto AD=BC\:(opposite\: sides\:are\: equal\:in\:p||gram\:ABCD)}$

$\rm{\leadsto \angle\:ADP=\angle\:CBQ\:(BQ\:and\:DP\: bisect\:\angle\:ADC\:and\:\angle\:ABC\:in\:p||gram)}$

$\rm{\leadsto \therefore\:\triangle\:ADP\cong\:\triangle\:BQC\:(by\:A.S.A\:criterian\:of\: Congruency)}$

$\tt{\leadsto \therefore\:DP=BQ(by\:CPCTC)}$

$\rm\small{\leadsto Hence,\:DP=BQ\: proved}$

Answer 2

Given :-

ABCD is a parallelogram. P and Q are points  on the diagonal AC such that DP and BQ are  perpendiculars to AC.

To Prove :-

DP = BQ

Solution :-

Let the two triangle be

ΔADP and ΔBQC

Now

∠ADP will be of 90°

Also,

∠ADP = ∠BQC

90 = 90

According to the property of a parallelogram. Opposite sides of a parallelogram will be equal.

AD and BC  is the opposite side of the parallelogram

AD = BC

∠ADP = ∠CBQ

By using ASA (Angle-Side-Angle) property

ΔADP ≅ ΔBQC

By using CPCTC property.

DP = BQ

Know More :-

ASA property - Angle side angle property is the property in which the side and the angle between the side are always congurent.

CPCTC property -  It says that corrosponding part of the congurent of the triangle are always congurent to each other.

SSS property - Side-Side-Side property. All threee side of triangle are congurent to each other

SAS property - Side angle Side property. Property in which the two side and the angle formed by them are congurent to each other

AAS - Angle Angle Side property. Two angle and a side of other shape are congurent