Question

H3: In △ABC, ∠B = 90◦.
BD is the bisector of ∠B with D on AC.
P and Q are the feet of altitudes from D on AB and BC respec- tively.
Prove that 2BPDQ is a square.
A
P
D
H6: In △ABC, A = 90◦. AD is altitude, −−→
D∈BC. LetAX bisect∠CAD. X∈ CD. show that △ABX is isosceles trian- gle.
BDXC
X
H7: Given ∠XY Z. Point P lies on bisector −−→
of ∠XYZ points Q,R are on YX and YZ respectively, such that P Q = P R. Show that Y Y Q = Y R.
P

Answer 1

Answer:

Given △ABC in which BD is the bisector of ∠B and a line PQ||AC meets AB,BC and BD at P,Q and R respectively.

Proof (i)

In △BQP, BR is the bisector of ∠B.

∴

BP

BQ

=

PR

QR

⇒ BQ.PR=BP.QR

⇒ PR.BQ=QR.BP [Hence proved]

(ii) In △ABC, we have

PQ∣∣AC [Given]

⇒

AP

AB

=

CQ

CB

[By Thale's Theorem]

⇒ AB×CQ=BC.AP [Hence proved