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Answer the Question :-
In the figure, AD is a median and P is a point on it. Which of the following is correct?
(i) ar(ΔAPB) = ar(ΔBPD)
(ii) ar(ΔAPB) = ar(ΔAPC)
(iii) ar(ΔAPB) = ar(ΔPDC)
Answers
Answer: OPTION B
We are given three options and we need to verify which of them are correct .
(i) Ar ( Δ APB ) = Ar ( Δ BPD )
In Δ APB and Δ BPD ,
There is no proof of congruency .
(ii) Ar( Δ APB ) = Ar ( Δ APC )
In Δ APB and Δ APC ,
∠PAB = ∠PAC
[ Median bisects the angles ]
AP = AP [ Common ]
BD : DC = AB : AC [ By angle bisector theorem ]
Since BD : DC = 1
AB : AC = 1
⇒ AB/AC = 1
⇒ AB = AC
Δ ABP ≅ Δ APC [ S.A.S ]
Hence Ar ( Δ ABP ) ≅ Ar ( ΔAPC )
(iii) Ar( Δ APB ) = Ar( Δ PDC )
No congruency .
Step-by-step explanation:
Whenever two triangles are congruent , their areas have to be equal .
This is why we have said that Ar( Δ APB ) = Ar ( Δ APC ) because the areas are equal .
The angle bisector theorem is used in the above problem .
The congruency proof is according to the S.A.S rule .
S.A.S rule states that 2 triangles are congruent if 2 sides and the including angle is equal .
Given
=>AD is a median and P is a point on traingle ABC.
Now,we have to prove the following conditions if they are true.
1)ar(ΔAPB) = ar(ΔBPD)
There is no such way to prove that they are congruent and hence their ratio of areas cant be determined,so we cant prove that they have same areas.
2)ar(ΔAPB) = ar(ΔAPC)
Since we know that =>
Area of ABD = Area of ADC
(since Median AD divides the traingle in two equal parts). --------(i)
Also,area of BPD = Area of PDC
(Since Median AD divides the traingle in two equal parts). -----(ii)
hence, substracting (ii) from (i) we get =>
ar(ABD) - ar(BPD) = ar(ADC) - ar(PDC)
=>Area of APB = Area of APC
iii) ar(ΔAPB) = ar(ΔPDC)
There are no such methord to prove that the area of traingles are equal it n according to the provided information.
EXTRA INFORMATION
Methords of congruency of two traingles=>
1)SAS ( SIDE, ANGLE,SIDE)
If two sides and one angle of one traingle is congruent with other traingle then they are congruent.
2)SSS(SIDE, SIDE, SIDE)
If three sides of a traingle is congruent with other 3 sides of another traingle,then they are congruent
3)ASA(ANGLE, SIDE ,ANGLE)
If two angles and one side of one traingle is congruent with other traingle then they are congruent.
4)AAS(ANGLE ,ANGLE ,SIDE)
If two angles and the non-included side are equal then the two traingles are congruent.
5)RHS(RIGHT ANGLE HYPOTENUSE SIDE)
If the hypotenuse and one side of one triangle are respectively equal to the hypotenuse and the corresponding side of the other triangle then they are congruent.