Question

In the given figure, D, E and F are the mid points of PQ, PR
and QR respectively and PG 1 QR. Prove that DEFQ is a
cyclic quadrilateral​

Answer 1

Given :- In the given figure, D, E and F are the mid points of PQ, PR and QR respectively and PG ⟂ QR.

To Prove :- DEFG is a cyclic quadrilateral .

Answer :-

Construction :- Join DG and DF .

in ∆PQR, we have,

→ D and E are mid points of PQ and PR .

so,

→ DE ll QR and DE = (1/2)QR --------- Eqn.(1)

similarly,

→ EF || PQ and EF = (1/2)PQ --------- Eqn.(2)

→ DF ll PR and DF = (1/2)PR --------- Eqn.(3)

also,

• PD = DQ = (1/2)PQ .
• PE = ER = (1/2)PR .
• QF = FR = (1/2)QR .

then, from Eqn.(1) and Eqn.(2)

→ DE ll QR and EF ll PQ .

• DEFQ is a parallelogram .

similarly, from Eqn.(2) and Eqn.(3)

→ DF ll PR and EF ll PQ .

• PDFE is a parallelogram .

so,

→ ∠DPE = ∠DFE (opposite angles of parallelogram PDFE are equal.)

→ ∠PRF = ∠DFQ (corresponding angles.)

→ ∠PDE = ∠PQR (corresponding angles.) ------- Eqn.(4)

then,

→ ∠GFE = ∠DFQ + ∠DFE

→ ∠GFE = ∠PRF + ∠DPE ----------- Eqn.(5)

now, in PQG, Let DE cuts PG at O .

→ DO ll QG and DO = (1/2) QG .

so,

→ PO = OG .

then,

→ ∠POD = ∠PGQ = 90° (corresponding angles.)

now, in ∆POD and ∆GOD,

→ ∠POD = ∠DOG (each 90°)

→ OD = OD (common)

→ PO = OG (DO = (1/2) QG )

so,

→ ∆POD ~ ∆GOD (By SAS.)

then,

→ ∠PDO = ∠GDO (By CPCT.)

from Eqn.(4)

→ ∠PDE = ∠PQR

→ ∠PDO = ∠PQR

or,

→ ∠GDO = ∠PQR --------- Eqn.(6)

now, in quadrilateral DGFE,

→ ∠GDE + ∠GFE = ∠GDE + ∠PRF + ∠DPE (using Eqn.5)

→ ∠GDE + ∠GFE = ∠PQR + ∠PRF + ∠DPE (using Eqn.6)

→ ∠GDE + ∠GFE = ∠PQR + ∠PRQ + ∠DPE

→ ∠GDE + ∠GFE = ∠PQR + ∠PRQ + ∠QPR

→ ∠GDE + ∠GFE = 180° (Angle sum property.)

since sum of opposite angles of quadrilateral DGFE is equal to 180° .

therefore, we can conclude that, DEFG is a cyclic quadrilateral .

Learn more :-

in triangle ABC seg DE parallel side BC. If 2 area of triangle ADE = area of quadrilateral DBCE find AB : AD show that B...

https://brainly.in/question/15942930

2) In ∆ABC seg MN || side AC, seg MN divides ∆ABC into two parts of equal area. Determine the value of AM / AB

https://brainly.in/question/37634605