To determine the ratio in which the line segment joining the points [tex]\((2, -4)\)[/tex] and [tex]\((5, 8)\)[/tex] is divided by the [tex]\(x\)[/tex]-axis, follow these steps:
1. Identify the coordinates of the given points:
- Point [tex]\(A\)[/tex] is [tex]\((2, -4)\)[/tex]
- Point [tex]\(B\)[/tex] is [tex]\((5, 8)\)[/tex]
2. Determine the coordinates where the line segment intersects the [tex]\(x\)[/tex]-axis:
- Let the coordinates of the intersection point on the [tex]\(x\)[/tex]-axis be [tex]\((x, 0)\)[/tex].
3. Use the section formula to find the ratio:
- The section formula states that if a point [tex]\((P_x, P_y)\)[/tex] divides a line segment joining [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the ratio [tex]\(m:n\)[/tex], then [tex]\(P_x = \frac{mx_2 + nx_1}{m+n}\)[/tex] and [tex]\(P_y = \frac{my_2 + ny_1}{m+n}\)[/tex].
4. Set up the equation for the [tex]\(y\)[/tex]-coordinate:
- Since the [tex]\(x\)[/tex]-axis has [tex]\(y = 0\)[/tex],
- Use the [tex]\(y\)[/tex]-coordinate section formula: [tex]\[ 0 = \frac{m \cdot 8 + n \cdot (-4)}{m + n} \][/tex]
5. Solve for the ratio [tex]\(m:n\)[/tex]:
- Simplify the equation: [tex]\[ 0 = 8m - 4n \][/tex]
- This implies [tex]\(8m = 4n\)[/tex].
6. Find the relationship between [tex]\(m\)[/tex] and [tex]\(n\)[/tex]:
- Divide both sides by 4: [tex]\[ 2m = n \][/tex]
- Hence, [tex]\( \frac{m}{n} = \frac{1}{2} \)[/tex].
So, the ratio in which the line segment joining the points [tex]\((2, -4)\)[/tex] and [tex]\((5, 8)\)[/tex] is divided by the [tex]\(x\)[/tex]-axis is [tex]\(\frac{1}{2}\)[/tex], or equivalently [tex]\(1:2\)[/tex].
To put it explicitly, the division of the segment by the [tex]\(x\)[/tex]-axis is:
- Ratio [tex]\(m:n = \frac{1}{2}\)[/tex]
- Values [tex]\(m = 4\)[/tex], [tex]\(n = 8\)[/tex]
Hence, the line segment is divided in the ratio [tex]\(1:2\)[/tex] with specific values [tex]\(4\)[/tex] and [tex]\(8\)[/tex].
