the sum of the perpendicular drawn from an Interior point of an equilateral triangle is 20 cm what is the length of side of the triangle find???
Answers
Answer:
Step-by-step explanation:
Given: an equilateral triangle ABC with side a, an interior point P and the perpendiculars PD, PE, PF respectively on to the sides BC, CA, AB.
To prove: PD+PE+PF is independent of the point P, but depends only on △ABC.
Observe
area(ABC)= area(PBC)+ area(PCA)+ area(PAB).
But we know
area(PBC)=
2
1
BC×PD,
area(PCA)=
2
1
CA×PE,
area(PAB)=
2
1
AB×PF.
But we know BC=CA=AB=a, since ABC is equilateral. Hence
area(ABC)=
2
1
a(PD+PE+PF).
We obtain
PD+PE+PF=
a
2area(ABC)
.
Hence PD+PE+PF depends only on the triangle ABC, but not on the point P.
an equilateral triangle ABC with side a, an interior point P and the perpendiculars PD, PE, PF respectively on to the sides BC, CA, AB.
To prove: PD+PE+PF is independent of the point P, but depends only on △ABC.
Observe
area(ABC)= area(PBC)+ area(PCA)+ area(PAB).
But we know
area(PBC)=
2
1
BC×PD,
area(PCA)=
2
1
CA×PE,
area(PAB)=
2
1
AB×PF.
But we know BC=CA=AB=a, since ABC is equilateral. Hence
area(ABC)=
2
1
a(PD+PE+PF).
We obtain
PD+PE+PF=
a
2area(ABC)
.
Hence PD+PE+PF depends only on the triangle ABC, but not on the point P