Question

a point p(1,0) divides line segment ab in a ratio of 3:5 from endpoint a(4,3). determine the coordinates of point b.

Explanation:

Step1: Use section - formula for x - coordinate

Let \(A(x_1,y_1)=(4,3)\), \(P(x,y)=(1,0)\) and the ratio \(m:n = 3:5\). The section - formula for the x - coordinate of a point \(P\) that divides the line segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\) is \(x=\frac{mx_2+nx_1}{m + n}\). Substitute \(x = 1\), \(x_1=4\), \(m = 3\) and \(n = 5\) into the formula: \(1=\frac{3x_2+5\times4}{3 + 5}\).

$$1=\frac{3x_2 + 20}{8}$$

$$8=3x_2+20$$

$$3x_2=8 - 20=-12$$

$$x_2=-4$$

Step2: Use section - formula for y - coordinate

The section - formula for the y - coordinate of a point \(P\) that divides the line segment joining \(A(x_1,y_1)\) and \(B(x_2,y_2)\) in the ratio \(m:n\) is \(y=\frac{my_2+ny_1}{m + n}\). Substitute \(y = 0\), \(y_1 = 3\), \(m = 3\) and \(n = 5\) into the formula: \(0=\frac{3y_2+5\times3}{3 + 5}\).

$$0=\frac{3y_2+15}{8}$$

$$0=3y_2+15$$

$$3y_2=-15$$

$$y_2=-5$$

Answer:

\((-4,-5)\)