P is the mid-point of the side CD of a parallelogram ABCD. A line through C parallel to PA intersects AB at Q and DA produced at R. Prove that DA = AR and CQ = QR.
Solution:
Given, ABCD is a parallelogram
P is the midpoint of the side CD
A line through C parallel to PA intersects AB at Q and DA produced at R.
We have to prove that DA = AR and CQ = QR
We know that the opposite sides of a parallelogram are parallel and congruent.
BC || AD and BC = AD
AB || CD and AB = CD
Since P is the midpoint of DC.
DP = PC = 1/2 CD ----------- (1)
Given, QC || AP
PC || AQ
Therefore, APCQ is a parallelogram
So, AQ = PC
From (1),
AQ = PC = 1/2 CD
Since AB = CD
PC = 1/2 AB = BQ
Considering triangles AQR and BQC,
AQ = BQ
We know that the vertically opposite angles are equal
∠AQR = ∠BQC
We know that the alternate interior angles are equal
∠ARQ = ∠BCQ
By ASA criterion, the triangles AQR and BQC
The Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem states that when two triangles are congruent, then their corresponding sides and angles are also congruent or equal in measurements.
By CPCTC,
AR = BC
Given, BC = AD
So, AR = AD
By CPCTC,
CQ =QR
Therefore, it is proved that AR = AD and CQ = QR.
✦ Try This: Diagonals PR and QS of a rhombus PQRS are 20 cm and 48 cm respectively. Find the length of side PQ.
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 8
NCERT Exemplar Class 9 Maths Exercise 8.4 Problem 18
P is the mid-point of the side CD of a parallelogram ABCD. A line through C parallel to PA intersects AB at Q and DA produced at R. Prove that DA = AR and CQ = QR
Summary:
P is the mid-point of the side CD of a parallelogram ABCD. A line through C parallel to PA intersects AB at Q and DA produced at R. It is proven that DA = AR and CQ = QR
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