You have studied in Class IX, that a median of a triangle divides it into two triangles of equal areas. Verify this result for ΔABC whose vertices are A(4,−6),B(3,−2) and C(5,2).

Question

You have studied in Class IX, that a median of a triangle divides it into two triangles of equal areas. Verify this result for ΔABC whose vertices are A(4,−6),B(3,−2) and C(5,2).

Filo · Accepted Answer

Let AD is the median of 	riangle ABC. Then D is the midpoint of BC.	hereforeCo-ordinates of D=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}ight)=\left(\dfrac{3+5}{2},\dfrac{-2+2}{2}ight)=(4,0)Median AD divide the 	riangle ABC into two triangles.	herefore Area of 	riangle ABD=\dfrac{1}{2}\left[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)ight]Here(x_1,y_1)=(4,-6)(x_2,y_2)=(3,-2)(x_3,y_3)=(4,0)Area of 	riangle ABD=\dfrac{1}{2}\left[4(-2-0)+3(0+6)+4(-6+2)ight]= \dfrac{1}{2}\left[-8+18-16 ight]=\dfrac{-6}{2}=3	herefore  Area of 	riangle ADC=\dfrac{1}{2}\left[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)ight]Here(x_1,y_1)=(4,-6)(x_2,y_2)=(4,0)(x_3,y_3)=(5,-2)Area of 	riangle ADC=\dfrac{1}{2}\left[4(0+2)+4(-2+6)+5(-6-0)ight]= \dfrac{1}{2}\left[-8+16-30 ight]=\dfrac{-6}{2}=3	herefore  Area of 	riangle ABD=Area of 	riangle ADCHence it is proved that a median of a triangle divides it into two triangles of equal areas.

Question Text	You have studied in Class IX, that a median of a triangle divides it into two triangles of equal areas. Verify this result for $Δ A BC$ whose vertices are $A (4, - 6), B (3, - 2)$ and $C (5, 2)$ .
Answer Type	Text solution:1
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You have studied in Class IX, that a median of a triangle divides it into two triangles of equal areas. Verify this result for $Δ A BC$ whose vertices are $A (4, - 6), B (3, - 2)$ and $C (5, 2)$ .

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