Answer :

Given: ABC is a triangle, PQ || BC; AD is the median which cuts PQ at R.

To prove: AD bisects PQ at R.

Proof: In Δ ABD; PR || BD

PB RD

In Δ ACD, RQ || DC

∴

RD QC

In ΔAPR and ΔABD,

∠APR = ∠ABD (corresponding angles.)

∠ARP = ∠ADB (corresponding angles.)

∴ Δ APR is similar to Δ ABD (AA similarity)

∴

AB AD BD

Similarly Δ ARQ is similar to Δ ADC

\\\\

AC AD DC

According to equation (i) and (ii),

AD BD DC

but BD = DC (given)

∴ PR = RQ

or AD bisects PQ at R (proved).

To prove: AD bisects PQ at R.

Proof: In Δ ABD; PR || BD

__AP__=__AR__(BPT)PB RD

In Δ ACD, RQ || DC

∴

__AR__=__AQ__(BPT)RD QC

In ΔAPR and ΔABD,

∠APR = ∠ABD (corresponding angles.)

∠ARP = ∠ADB (corresponding angles.)

∴ Δ APR is similar to Δ ABD (AA similarity)

∴

__AP__=__AR__=__PR__(corresponding sides of similar triangles are proportional)----(i)AB AD BD

Similarly Δ ARQ is similar to Δ ADC

\\\\

__AQ__=__AR__=__RQ__-----(ii)AC AD DC

According to equation (i) and (ii),

__AR__=__PR__=__RQ__AD BD DC

but BD = DC (given)

∴ PR = RQ

or AD bisects PQ at R (proved).

Rate this question :

How useful is this solution?

We strive to provide quality solutions. Please rate us to serve you better.

Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Expertsview all courses

Dedicated counsellor for each student

24X7 Doubt Resolution

Daily Report Card

Detailed Performance Evaluation

RELATED QUESTIONS :

Draw the graph ofNCERT - Mathematics