Question

select the correct answer. in the diagram, the areas of $delta adc$ and $delta dcb$ 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)

Explanation:

Step1: Recall the section - formula concept

Since the ratio of the areas of $\triangle ADC$ and $\triangle DCB$ is $3:4$, and they have the same height (the perpendicular distance from $D$ to $AB$), the ratio of the lengths of $AC$ and $CB$ is also $3:4$ (because $Area=\frac{1}{2}\times base\times height$).

Step2: Use the section - formula

The section - formula for a point $C(x,y)$ that divides the line - segment joining $A(x_1,y_1)$ and $B(x_2,y_2)$ in the ratio $m:n$ is given by $x=\frac{mx_2+nx_1}{m + n}$ and $y=\frac{my_2+ny_1}{m + n}$. Here, $x_1 = 1,y_1=-9,x_2 = 8,y_2=-2,m = 3,n = 4$.
For the $x$ - coordinate of $C$:
$x=\frac{3\times8 + 4\times1}{3 + 4}=\frac{24+4}{7}=\frac{28}{7}=4$
For the $y$ - coordinate of $C$:
$y=\frac{3\times(-2)+4\times(-9)}{3 + 4}=\frac{-6-36}{7}=\frac{-42}{7}=-6$

Answer:

D. $(4,-6)$