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1. Find the coordinates of a point which divides the line segment joining [tex]\((x, y)\)[/tex] and [tex]\((0,0)\)[/tex] in the ratio of [tex]\(m_1: m_2\)[/tex] externally.

2. Find the coordinates of the midpoint of a line segment joining the points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex].

3. Find the coordinates of a point which divides the line segment [tex]\(AB\)[/tex] in the given ratio:

a. [tex]\(A(1,1)\)[/tex] and [tex]\(B(4,4)\)[/tex]; Ratio [tex]\(= 1:2\)[/tex]
b. [tex]\(A(2,6)\)[/tex] and [tex]\(B(8,3)\)[/tex]; Ratio [tex]\(= 2:1\)[/tex]
c. [tex]\(A(4,5)\)[/tex] and [tex]\(B(12,1)\)[/tex]; Ratio [tex]\(= 3:1\)[/tex]
d. [tex]\(A(2,6)\)[/tex] and [tex]\(B(12,1)\)[/tex]; Ratio [tex]\(= 3:2\)[/tex]
e. [tex]\(A(-7,5)\)[/tex] and [tex]\(B(3,0)\)[/tex]; Ratio [tex]\(= 3:2\)[/tex]
f. [tex]\(A(-7,5)\)[/tex] and [tex]\(B(5,-1)\)[/tex]; Ratio [tex]\(= 5:1\)[/tex]

4. Find the ratio in which the point [tex]\(P\)[/tex] divides [tex]\(AB\)[/tex] internally:

a. [tex]\(A(1,1)\)[/tex], [tex]\(B(4,4)\)[/tex], and [tex]\(P(2,2)\)[/tex]
b. [tex]\(A(2,6)\)[/tex], [tex]\(B(8,3)\)[/tex], and [tex]\(P(6,4)\)[/tex]
c. [tex]\(A(-3,3)\)[/tex], [tex]\(B(5,-1)\)[/tex], and [tex]\(P(3,0)\)[/tex]
d. [tex]\(A(-3,3)\)[/tex], [tex]\(B(5,-1)\)[/tex], and [tex]\(P(-1,2)\)[/tex]
e. [tex]\(A(2,6)\)[/tex], [tex]\(B(12,1)\)[/tex], and [tex]\(P(10,2)\)[/tex]
f. [tex]\(A(2,6)\)[/tex], [tex]\(B(14,0)\)[/tex], and [tex]\(P(12,1)\)[/tex]

Sagot :

GinnyAnsweridn
To find the coordinates of a point that divides a line segment [tex]\( AB \)[/tex] in a given ratio [tex]\( m_1 : m_2 \)[/tex], we can use the section formula. For a line segment joining points [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex], the coordinates of the point that divides the line segment in the ratio [tex]\( m_1 : m_2 \)[/tex] are given by:

[tex]\[ \left( \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \frac{m_1 y_2 + m_2 y_1}{m_1 + m_2} \right) \][/tex]

Let's apply this formula step-by-step for each given pair of points and their respective ratios:

### (a) [tex]\( A (1, 1) \)[/tex] and [tex]\( B (4, 4) \)[/tex]; Ratio [tex]\( = 1: 2 \)[/tex]
- [tex]\( x_1 = 1 \)[/tex], [tex]\( y_1 = 1 \)[/tex], [tex]\( x_2 = 4 \)[/tex], [tex]\( y_2 = 4 \)[/tex]
- [tex]\( m_1 = 1 \)[/tex], [tex]\( m_2 = 2 \)[/tex]

The coordinates of the point dividing the line segment are:
[tex]\[ \left( \frac{1 \cdot 4 + 2 \cdot 1}{1 + 2}, \frac{1 \cdot 4 + 2 \cdot 1}{1 + 2} \right) = \left( \frac{4 + 2}{3}, \frac{4 + 2}{3} \right) = \left( 2, 2 \right) \][/tex]

### (b) [tex]\( A (2, 6) \)[/tex] and [tex]\( B (8, 3) \)[/tex]; Ratio [tex]\( = 2: 1 \)[/tex]
- [tex]\( x_1 = 2 \)[/tex], [tex]\( y_1 = 6 \)[/tex], [tex]\( x_2 = 8 \)[/tex], [tex]\( y_2 = 3 \)[/tex]
- [tex]\( m_1 = 2 \)[/tex], [tex]\( m_2 = 1 \)[/tex]

The coordinates of the point dividing the line segment are:
[tex]\[ \left( \frac{2 \cdot 8 + 1 \cdot 2}{2 + 1}, \frac{2 \cdot 3 + 1 \cdot 6}{2 + 1} \right) = \left( \frac{16 + 2}{3}, \frac{6 + 6}{3} \right) = \left( 6, 4 \right) \][/tex]

### (c) [tex]\( A (4, 5) \)[/tex] and [tex]\( B (12, 1) \)[/tex]; Ratio [tex]\( = 3: 1 \)[/tex]
- [tex]\( x_1 = 4 \)[/tex], [tex]\( y_1 = 5 \)[/tex], [tex]\( x_2 = 12 \)[/tex], [tex]\( y_2 = 1 \)[/tex]
- [tex]\( m_1 = 3 \)[/tex], [tex]\( m_2 = 1 \)[/tex]

The coordinates of the point dividing the line segment are:
[tex]\[ \left( \frac{3 \cdot 12 + 1 \cdot 4}{3 + 1}, \frac{3 \cdot 1 + 1 \cdot 5}{3 + 1} \right) = \left( \frac{36 + 4}{4}, \frac{3 + 5}{4} \right) = \left( 10, 2 \right) \][/tex]

### (d) [tex]\( A (2, 6) \)[/tex] and [tex]\( B (12, 1) \)[/tex]; Ratio [tex]\( = 3: 2 \)[/tex]
- [tex]\( x_1 = 2 \)[/tex], [tex]\( y_1 = 6 \)[/tex], [tex]\( x_2 = 12 \)[/tex], [tex]\( y_2 = 1 \)[/tex]
- [tex]\( m_1 = 3 \)[/tex], [tex]\( m_2 = 2 \)[/tex]

The coordinates of the point dividing the line segment are:
[tex]\[ \left( \frac{3 \cdot 12 + 2 \cdot 2}{3 + 2}, \frac{3 \cdot 1 + 2 \cdot 6}{3 + 2} \right) = \left( \frac{36 + 4}{5}, \frac{3 + 12}{5} \right) = \left( 8, 3 \right) \][/tex]

### (e) [tex]\( A (-7, 5) \)[/tex] and [tex]\( B (3, 0) \)[/tex]; Ratio [tex]\( = 3: 2 \)[/tex]
- [tex]\( x_1 = -7 \)[/tex], [tex]\( y_1 = 5 \)[/tex], [tex]\( x_2 = 3 \)[/tex], [tex]\( y_2 = 0 \)[/tex]
- [tex]\( m_1 = 3 \)[/tex], [tex]\( m_2 = 2 \)[/tex]

The coordinates of the point dividing the line segment are:
[tex]\[ \left( \frac{3 \cdot 3 + 2 \cdot -7}{3 + 2}, \frac{3 \cdot 0 + 2 \cdot 5}{3 + 2} \right) = \left( \frac{9 - 14}{5}, \frac{0 + 10}{5} \right) = \left( -1, 2 \right) \][/tex]

### (f) [tex]\( A (-7, 5) \)[/tex] and [tex]\( B (5, -1) \)[/tex]; Ratio [tex]\( = 5: 1 \)[/tex]
- [tex]\( x_1 = -7 \)[/tex], [tex]\( y_1 = 5 \)[/tex], [tex]\( x_2 = 5 \)[/tex], [tex]\( y_2 = -1 \)[/tex]
- [tex]\( m_1 = 5 \)[/tex], [tex]\( m_2 = 1 \)[/tex]

The coordinates of the point dividing the line segment are:
[tex]\[ \left( \frac{5 \cdot 5 + 1 \cdot -7}{5 + 1}, \frac{5 \cdot -1 + 1 \cdot 5}{5 + 1} \right) = \left( \frac{25 - 7}{6}, \frac{-5 + 5}{6} \right) = \left( 3, 0 \right) \][/tex]

This gives us the coordinates for each case in the problem statement.
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