Answer :
Sure! Let's break this problem into two parts:
1. Find the point on the line joining the points (3, 9) and (4, 5) whose y-coordinate is -3.
2. Find the points of trisection of the line segment joining (3, 9) and (4, 5).
### Part 1: Find the point with y-coordinate -3
First, we need to find the equation of the line passing through the points (3, 9) and (4, 5).
1. Calculate the slope (m):
The formula for the slope [tex]\(m\)[/tex] of a line passing through the points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the given points [tex]\((3, 9)\)[/tex] and [tex]\((4, 5)\)[/tex]:
[tex]\[ m = \frac{5 - 9}{4 - 3} = \frac{-4}{1} = -4 \][/tex]
2. Find the y-intercept (c):
The equation of the line can be written in the form [tex]\(y = mx + c\)[/tex].
To find [tex]\(c\)[/tex], we use one of the given points. Let's use [tex]\((3, 9)\)[/tex]:
[tex]\[ 9 = (-4) \cdot 3 + c \implies 9 = -12 + c \implies c = 21 \][/tex]
So, the equation of the line is:
[tex]\[ y = -4x + 21 \][/tex]
3. Find the x-coordinate when [tex]\(y = -3\)[/tex]:
Substitute [tex]\(y = -3\)[/tex] into the equation:
[tex]\[ -3 = -4x + 21 \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ -3 - 21 = -4x \implies -24 = -4x \implies x = \frac{-24}{-4} = 6 \][/tex]
Thus, the point on the line whose y-coordinate is -3 is:
[tex]\[ (6, -3) \][/tex]
### Part 2: Find the points of trisection of the line segment
Trisection points divide the segment into three equal parts. Let the points be [tex]\((x_1, y_1) = (3, 9)\)[/tex] and [tex]\((x_2, y_2) = (4, 5)\)[/tex].
1. Calculate the coordinates of the trisection points:
Let the trisection points be [tex]\(A\)[/tex] and [tex]\(B\)[/tex]. The coordinates [tex]\(A\)[/tex] and [tex]\(B\)[/tex] can be found using the section formula in the ratio 1:2 and 2:1.
For point [tex]\(A\)[/tex] (dividing the line in the ratio 1:2), the coordinates are:
[tex]\[ A = \left( \frac{2x_1 + x_2}{3}, \frac{2y_1 + y_2}{3} \right) \][/tex]
Substituting the values:
[tex]\[ A = \left( \frac{2 \cdot 3 + 4}{3}, \frac{2 \cdot 9 + 5}{3} \right) = \left( \frac{6 + 4}{3}, \frac{18 + 5}{3} \right) = \left( \frac{10}{3}, \frac{23}{3} \right) \approx (3.33, 7.67) \][/tex]
For point [tex]\(B\)[/tex] (dividing the line in the ratio 2:1), the coordinates are:
[tex]\[ B = \left( \frac{x_1 + 2x_2}{3}, \frac{y_1 + 2y_2}{3} \right) \][/tex]
Substituting the values:
[tex]\[ B = \left( \frac{3 + 2 \cdot 4}{3}, \frac{9 + 2 \cdot 5}{3} \right) = \left( \frac{3 + 8}{3}, \frac{9 + 10}{3} \right) = \left( \frac{11}{3}, \frac{19}{3} \right) \approx (3.67, 6.33) \][/tex]
Therefore, the trisection points of the line segment joining the points (3, 9) and (4, 5) are:
[tex]\[ (3.33, 7.67) \quad \text{and} \quad (3.67, 6.33) \][/tex]
### Summary
- The point on the line joining (3, 9) and (4, 5) whose y-coordinate is -3 is:
[tex]\[ (6, -3) \][/tex]
- The trisection points of the line segment joining (3, 9) and (4, 5) are:
[tex]\[ (3.33, 7.67) \quad \text{and} \quad (3.67, 6.33) \][/tex]
1. Find the point on the line joining the points (3, 9) and (4, 5) whose y-coordinate is -3.
2. Find the points of trisection of the line segment joining (3, 9) and (4, 5).
### Part 1: Find the point with y-coordinate -3
First, we need to find the equation of the line passing through the points (3, 9) and (4, 5).
1. Calculate the slope (m):
The formula for the slope [tex]\(m\)[/tex] of a line passing through the points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the given points [tex]\((3, 9)\)[/tex] and [tex]\((4, 5)\)[/tex]:
[tex]\[ m = \frac{5 - 9}{4 - 3} = \frac{-4}{1} = -4 \][/tex]
2. Find the y-intercept (c):
The equation of the line can be written in the form [tex]\(y = mx + c\)[/tex].
To find [tex]\(c\)[/tex], we use one of the given points. Let's use [tex]\((3, 9)\)[/tex]:
[tex]\[ 9 = (-4) \cdot 3 + c \implies 9 = -12 + c \implies c = 21 \][/tex]
So, the equation of the line is:
[tex]\[ y = -4x + 21 \][/tex]
3. Find the x-coordinate when [tex]\(y = -3\)[/tex]:
Substitute [tex]\(y = -3\)[/tex] into the equation:
[tex]\[ -3 = -4x + 21 \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ -3 - 21 = -4x \implies -24 = -4x \implies x = \frac{-24}{-4} = 6 \][/tex]
Thus, the point on the line whose y-coordinate is -3 is:
[tex]\[ (6, -3) \][/tex]
### Part 2: Find the points of trisection of the line segment
Trisection points divide the segment into three equal parts. Let the points be [tex]\((x_1, y_1) = (3, 9)\)[/tex] and [tex]\((x_2, y_2) = (4, 5)\)[/tex].
1. Calculate the coordinates of the trisection points:
Let the trisection points be [tex]\(A\)[/tex] and [tex]\(B\)[/tex]. The coordinates [tex]\(A\)[/tex] and [tex]\(B\)[/tex] can be found using the section formula in the ratio 1:2 and 2:1.
For point [tex]\(A\)[/tex] (dividing the line in the ratio 1:2), the coordinates are:
[tex]\[ A = \left( \frac{2x_1 + x_2}{3}, \frac{2y_1 + y_2}{3} \right) \][/tex]
Substituting the values:
[tex]\[ A = \left( \frac{2 \cdot 3 + 4}{3}, \frac{2 \cdot 9 + 5}{3} \right) = \left( \frac{6 + 4}{3}, \frac{18 + 5}{3} \right) = \left( \frac{10}{3}, \frac{23}{3} \right) \approx (3.33, 7.67) \][/tex]
For point [tex]\(B\)[/tex] (dividing the line in the ratio 2:1), the coordinates are:
[tex]\[ B = \left( \frac{x_1 + 2x_2}{3}, \frac{y_1 + 2y_2}{3} \right) \][/tex]
Substituting the values:
[tex]\[ B = \left( \frac{3 + 2 \cdot 4}{3}, \frac{9 + 2 \cdot 5}{3} \right) = \left( \frac{3 + 8}{3}, \frac{9 + 10}{3} \right) = \left( \frac{11}{3}, \frac{19}{3} \right) \approx (3.67, 6.33) \][/tex]
Therefore, the trisection points of the line segment joining the points (3, 9) and (4, 5) are:
[tex]\[ (3.33, 7.67) \quad \text{and} \quad (3.67, 6.33) \][/tex]
### Summary
- The point on the line joining (3, 9) and (4, 5) whose y-coordinate is -3 is:
[tex]\[ (6, -3) \][/tex]
- The trisection points of the line segment joining (3, 9) and (4, 5) are:
[tex]\[ (3.33, 7.67) \quad \text{and} \quad (3.67, 6.33) \][/tex]
