Explore Westonci.ca, the leading Q&A site where experts provide accurate and helpful answers to all your questions. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
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]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
