Question

In the given APQR, A and B are the mid-points of sides PQ and PR respectively. D and C are two points of QR such that AD BC. Prove that : quad ABCD=1/2 PQR​

Answer 1

Answer:

Here, we are joining A and C.

In ΔABC

P is the mid point of AB

Q is the mid point of BC

PQ∣∣AC [Line segments joining the mid points of two sides of a triangle is parallel to AC(third side) and also is half of it]

PQ=

2

1

AC

In ΔADC

R is mid point of CD

S is mid point of AD

RS∣∣AC [Line segments joining the mid points of two sides of a triangle is parallel to third side and also is half of it]

RS=

2

1

AC

So, PQ∣∣RS and PQ=RS [one pair of opposite side is parallel and equal]

In ΔAPS & ΔBPQ

AP=BP [P is the mid point of AB)

∠PAS=∠PBQ(All the angles of rectangle are 90

o

)

AS=BQ

∴ΔAPS≅ΔBPQ(SAS congruency)

∴PS=PQ

BS=PQ & PQ=RS (opposite sides of parallelogram is equal)

∴ PQ=RS=PS=RQ[All sides are equal]

∴ PQRS is a parallelogram with all sides equal

∴ So PQRS is a rhombus.

solution

Answer 2

Given: APQR, A and B are the mid-points of sides PQ and PR respectively. D and C are two points of QR such that AD BC

To Find: Proving that quad ABCD=1/2 QPR​

Solution:

In ΔABC

P is the mid point of AB and Q is the mid point of BC

Then, PQ∣∣AC

PQ= 1/2 AC

In ΔADC

R is the mid point of the CD line and S is the mid point of AD

RS∣∣AC

RS= 2 AC

So, PQ∣∣RS and PQ=RS

one pair of the opposite sides is parallel and equal.

In ΔAPS & ΔBPQ

AP=BP, P being the mid point of AB

∠PAS=∠PBQ

All the angles of the rectangle are 90 degrees

AS=BQ

∴ΔAPS≅ΔBPQ by the SAS congruency test.

∴PS=PQ

BS=PQ and  PQ=RS

opposite sides of a parallelogram are equal.

∴ PQ=RS=PS=RQ since all sides are equal.

∴ PQRS is a parallelogram with all sides equal.

Therefore, ABCD = 1/2 of AC

Where AC= PQ

This implies, that ABCD = 1/2 of PQ.

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