In given figure BO and CO are respectively the bisectors of - B and - C of ABC- AO produced meets BC at P show thati AB BP - AO OPii AC CP - AO OPiii AB AC - BP PCiv AP is the bisector of - BAC-

(i) In ABP , BO is the bisector of ∠ B ∴ #160; #160; AB BP = AO OP #160; #160; #160; [Hence proved] (ii) In ACP , ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard VII

Mathematics

Equal Angles Subtend Equal Sides

In given figu...

Question

In given figure BO and CO are respectively the bisectors of $∠ B$ and $∠ C$ of $△ A B C$ . AO produced meets BC at P show that

(i) $\frac{A B}{B P} = \frac{A O}{O P}$

(ii) $\frac{A C}{C P} = \frac{A O}{O P}$

(iii) $\frac{A B}{A C} = \frac{B P}{P C}$

(iv) AP is the bisector of $∠ B A C$ .

Open in App

Solution

(i) In $△ A B P$ , BO is the bisector of $∠ B$

$∴$ $\frac{A B}{B P} = \frac{A O}{O P}$ $[H e n c e p r o v e d]$

(ii) In $△ A C P$ , OC is the bisector of $∠ C$

$∴$ $\frac{A C}{C P} = \frac{A O}{O P}$ $[H e n c e p r o v e d]$

(iii) We have, proved that

$\frac{A B}{B P} = \frac{A O}{O P}$ and $\frac{A C}{C P} = \frac{A O}{O P}$

$\Rightarrow$ $\frac{A B}{B P} = \frac{A C}{C P}$

$\Rightarrow$ $\frac{A B}{A C} = \frac{B P}{P C}$ $[H e n c e p r o v e d]$

(iv) As proved above that in $△ A B C$ , we have

$\frac{A B}{A C} = \frac{B P}{C P}$

$\Rightarrow A P$ is the bisector of $∠ B A C$ .

Suggest Corrections

Similar questions

BO and CO are respectively the bisectors of angle B and C of ∆ABC. AO produced meets BC at P. Show that AO/BP = AO/OP , AC/CP = AO/OP ,

AB/AC = BP/PC , AP is the bisector of angle BAC.

BO and CO are respectively the bisectors of $∠ B$ and $∠ C$ of $Δ A B C$ . AO produced meets BC at P.

Show that [4 MARKS]

(i) $\frac{A B}{B P} = \frac{A O}{O P}$

(ii) $\frac{A C}{C P} = \frac{A O}{O P}$

(iii) $\frac{A B}{A C} = \frac{B P}{P C}$

(iv) $A P$ is the bisector of $∠ B A C$ .

Q. If

B O

and

C O

are respectively the fraction Of

∠ B

∠ C

△ A B C

A O

produced meets as at

P

and find the

A B / A C

In $Δ$ ABC, AB = AC and the bisectors of angles B and C intersect at point O. Prove that :

(i) BO = CO

(ii) AO bisects $∠ B A C$

Q. In a triangle ABC, AB = AC and the bisectors of angles B and C intersect at O. Prove that BO = CO and AO is the bisector of angle

∠ B A C

Join BYJU'S Learning Program

In given figure BO and CO are respectively the bisectors of ∠B and ∠C of △ABC. AO produced meets BC at P show that(i) ABBP=AOOP(ii) ACCP=AOOP(iii) ABAC=BPPC(iv) AP is the bisector of ∠BAC.