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QuestionPoint BB divides segment AC‾\overline{AC} in a 1:31:3 ratio. Given A(−4,−7)A(-4,-7) and B(12,5)B(12,5), find coordinates of CC.

Studdy Solution

STEP 1

Assumptions1. Point BB partitions line segment AC‾\overline{AC} in the ratio 1313. . The coordinates of point AA are (−4,−7)(-4,-7).
3. The coordinates of point BB are (12,5)(12,5).
4. We need to find the coordinates of point CC.

STEP 2

We can use the section formula to find the coordinates of point CC. The section formula is given byC(x,y)=(m1x2+m2x1m1+m2,m1y2+m2y1m1+m2)C(x,y) = \left( \frac{m1x2 + m2x1}{m1 + m2}, \frac{m1y2 + m2y1}{m1 + m2} \right)where m1:m2m1:m2 is the given ratio, and (x1,y1)(x1,y1) and (x2,y2)(x2,y2) are the coordinates of points AA and BB respectively.

STEP 3

Now, plug in the given values for the ratio and the coordinates of points AA and BB to calculate the coordinates of point CC.
C(x,y)=(1⋅(−)+3⋅121+3,1⋅(−7)+3⋅51+3)C(x,y) = \left( \frac{1 \cdot (-) +3 \cdot12}{1 +3}, \frac{1 \cdot (-7) +3 \cdot5}{1 +3} \right)

STEP 4

Calculate the x-coordinate of point CC.
Cx=1⋅(−4)+3⋅121+3=−4+364=324=8C_x = \frac{1 \cdot (-4) +3 \cdot12}{1 +3} = \frac{-4 +36}{4} = \frac{32}{4} =8

STEP 5

Calculate the y-coordinate of point CC.
Cy=1⋅(−7)+3⋅51+3=−7+154=84=2C_y = \frac{1 \cdot (-7) +3 \cdot5}{1 +3} = \frac{-7 +15}{4} = \frac{8}{4} =2

STEP 6

So, the coordinates of point CC are (8,2)(8,2).
The coordinates of point CC are (8,2)(8,2).

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