Question

drag the tiles to the boxes to form correct pairs. not all tiles will be used. point c divides $overline{ab}$ in a particular ratio. match point c and the ratio into which c divides $overline{ab}$ with the endpoints of $overline{ab}$. point c(-3.6, -3.4) divides $overline{ab}$ in the ratio 2:3. point c(4, 1.6) divides $overline{ab}$ in the ratio 3:2. point c(3.5, -2.5) divides $overline{ab}$ in the ratio 1:7. point c(8, 9) divides $overline{ab}$ in the ratio 5:3. point c(-2, 5) divides $overline{ab}$ in the ratio 2:6. point c(0, 1) divides $overline{ab}$ in the ratio 4:7. a(4, -3) and b(-7, 8) a(-5, 2) and b(7, 14) a(-2, -1) and b(-6, -7) a(3, 4) and b(11, 12)

Explanation:

Step1: Recall section - formula

If a point \(C(x,y)\) divides the line - segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\), then \(x=\frac{mx_2+nx_1}{m + n}\) and \(y=\frac{my_2+ny_1}{m + n}\).

Step2: Check each pair

For \(A(4,-3)\) and \(B(-7,8)\), \(m = 2\), \(n = 3\), \(x=\frac{2\times(-7)+3\times4}{2 + 3}=\frac{-14 + 12}{5}=-0.4\), \(y=\frac{2\times8+3\times(-3)}{2 + 3}=\frac{16-9}{5}=1.4\) (not a match).
For \(A(-5,2)\) and \(B(7,14)\), \(m = 3\), \(n = 2\), \(x=\frac{3\times7+2\times(-5)}{3 + 2}=\frac{21-10}{5}=\frac{11}{5}=2.2\) (not a match).
For \(A(-2,-1)\) and \(B(-6,-7)\), \(m = 1\), \(n = 7\), \(x=\frac{1\times(-6)+7\times(-2)}{1 + 7}=\frac{-6-14}{8}=\frac{-20}{8}=-2.5\) (not a match).
For \(A(3,4)\) and \(B(11,12)\), \(m = 5\), \(n = 3\), \(x=\frac{5\times11+3\times3}{5 + 3}=\frac{55 + 9}{8}=\frac{64}{8}=8\), \(y=\frac{5\times12+3\times4}{5 + 3}=\frac{60+12}{8}=\frac{72}{8}=9\) (match).
For \(A(4,-3)\) and \(B(-7,8)\), \(m = 4\), \(n = 7\), \(x=\frac{4\times(-7)+7\times4}{4 + 7}=\frac{-28 + 28}{11}=0\), \(y=\frac{4\times8+7\times(-3)}{4 + 7}=\frac{32-21}{11}=1\) (match).
For \(A(-5,2)\) and \(B(7,14)\), \(m = 2\), \(n = 6\), \(x=\frac{2\times7+6\times(-5)}{2 + 6}=\frac{14-30}{8}=\frac{-16}{8}=-2\), \(y=\frac{2\times14+6\times2}{2 + 6}=\frac{28 + 12}{8}=\frac{40}{8}=5\) (match).
For \(A(-2,-1)\) and \(B(-6,-7)\), \(m = 2\), \(n = 3\), \(x=\frac{2\times(-6)+3\times(-2)}{2 + 3}=\frac{-12-6}{5}=\frac{-18}{5}=-3.6\), \(y=\frac{2\times(-7)+3\times(-1)}{2 + 3}=\frac{-14-3}{5}=\frac{-17}{5}=-3.4\) (match).
For \(A(-5,2)\) and \(B(7,14)\), \(m = 1\), \(n = 7\), \(x=\frac{1\times7+7\times(-5)}{1 + 7}=\frac{7-35}{8}=\frac{-28}{8}=-3.5\) (not a match).

Answer:

Point \(C(-3.6,-3.4)\) divides \(\overline{AB}\) in the ratio \(2:3\) ↔ \(A(-2,-1)\) and \(B(-6,-7)\)
Point \(C(4,1.6)\) divides \(\overline{AB}\) in the ratio \(3:2\) ↔ No match
Point \(C(3.5,-2.5)\) divides \(\overline{AB}\) in the ratio \(1:7\) ↔ No match
Point \(C(8,9)\) divides \(\overline{AB}\) in the ratio \(5:3\) ↔ \(A(3,4)\) and \(B(11,12)\)
Point \(C(-2,5)\) divides \(\overline{AB}\) in the ratio \(2:6\) ↔ \(A(-5,2)\) and \(B(7,14)\)
Point \(C(0,1)\) divides \(\overline{AB}\) in the ratio \(4:7\) ↔ \(A(4,-3)\) and \(B(-7,8)\)