Question

Select the correct answer.

In the diagram, the areas of triangle abc and triangle dce are in a ratio of 3 : 4. What are the coordinates of point C?

A.
(3, -7)
B.
(5, -5)
C.
(7, -3)
D.
(4, -6)

Answer 1

The correct answer is

D. (4, -6)

Answer 2

Answer: The correct option is (D) (4, -6).

Step-by-step explanation: Given that the areas of the triangles ADC and DCB are in the ratio 3 : 4.

We are to find the co-ordinates of point C.

From the diagram, we note that

the co-ordinates of point A and B are A(1, -9) and B(8,-2).

So, the length of the line segment AB, calculated by distance formula, is

`AB=√((1-8)^2+(-9+2)^2)=√(49+49)=7\sqrt2.`

Now, area of ΔADC is

`A_(ABC)=(1)/(2)* AC* CD,`

and area of ΔDCB is

`A_(DCE)=(1)/(2)* BC* CD.`

According to the given information, we have

`A_(ADC):A_(DCB)=3:4\\\\\\\Rightarrow (A_(ADC))/(A_(DCE))=(3)/(4)\\\\\\\Rightarrow ((1)/(2)* AC* CD)/((1)/(2)* BC* CD)=(3)/(4)\\\\\\\Rightarrow (AC)/(BC)=3:4.`

So, the point C divides the line segment AB internally in the ratio 3 : 4.

We know that

if a point divides a line segment with end-points (a, b) and (c, d) internally in the ration m : n, then its co-ordinates are

`\left((mc+na)/(m+n),(md+nb)/(m+n)\right).`

Since point C divides the line segment AB with end-points A(1, -9) and B(8, -2) internally, so the co-ordinates of point C will be

`\left((3* 8+4* 1)/(3+4),(3* (-2)+4* (-9))/(3+4)\right)\\\\\\=\left((28)/(7),(-42)/(7)\right)\\\\\\=(4, -6).`

Thus, the co-ordinates of point C are (4, -6).

Option (D) is correct.