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Find the coordinates of a point that divides the line segment joining the points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the ratio [tex]\(m : n\)[/tex] externally.

1. Find the coordinates of a point that divides the line segment joining the points [tex]\((x, y)\)[/tex] and [tex]\((0,0)\)[/tex] in the ratio [tex]\(m_1: m_2\)[/tex] internally.
2. Find the coordinates of a point that divides the line segment joining the points [tex]\((x, y)\)[/tex] and [tex]\((0, \theta)\)[/tex] in the ratio [tex]\(m_1: m_2\)[/tex] externally.
3. 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].
4. Find the coordinates of the midpoint of a line segment joining the points [tex]\((a, b)\)[/tex] and [tex]\((0, 0)\)[/tex].

Solve the following:

1. 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]

2. 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]



Answer :

GinnyAnswer
### Solution

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

To find the coordinates [tex]\((x, y)\)[/tex] of a point that divides the line segment joining the points [tex]\(A(x_1, y_1)\)[/tex] and [tex]\(B(x_2, y_2)\)[/tex] in the ratio [tex]\(m : n\)[/tex], we use the section formula:

- Internal Division:
[tex]\[ x = \frac{m x_2 + n x_1}{m + n} \][/tex]
[tex]\[ y = \frac{m y_2 + n y_1}{m + n} \][/tex]

Let's solve each part:

(a) [tex]\( A(1, 1) \)[/tex] and [tex]\( B(4, 4) \)[/tex]; Ratio [tex]\(= 1 : 2\)[/tex]

Using the section formula for internal division:
[tex]\[ x = \frac{1 \cdot 4 + 2 \cdot 1}{1 + 2} = \frac{4 + 2}{3} = 2 \][/tex]
[tex]\[ y = \frac{1 \cdot 4 + 2 \cdot 1}{1 + 2} = \frac{4 + 2}{3} = 2 \][/tex]
So, the coordinates of the point that divides the line segment internally in the ratio [tex]\(1 : 2\)[/tex] are [tex]\((2.0, 2.0)\)[/tex].

(b) [tex]\( A(2, 6) \)[/tex] and [tex]\( B(8, 3) \)[/tex]; Ratio [tex]\(= 2 : 1\)[/tex]

Using the section formula for internal division:
[tex]\[ x = \frac{2 \cdot 8 + 1 \cdot 2}{2 + 1} = \frac{16 + 2}{3} = 6 \][/tex]
[tex]\[ y = \frac{2 \cdot 3 + 1 \cdot 6}{2 + 1} = \frac{6 + 6}{3} = 4 \][/tex]
So, the coordinates of the point that divides the line segment internally in the ratio [tex]\(2 : 1\)[/tex] are [tex]\((6.0, 4.0)\)[/tex].

(c) [tex]\( A(4, 5) \)[/tex] and [tex]\( B(12, 1) \)[/tex]; Ratio [tex]\(= 3 : 1\)[/tex]

Using the section formula for internal division:
[tex]\[ x = \frac{3 \cdot 12 + 1 \cdot 4}{3 + 1} = \frac{36 + 4}{4} = 10 \][/tex]
[tex]\[ y = \frac{3 \cdot 1 + 1 \cdot 5}{3 + 1} = \frac{3 + 5}{4} = 2 \][/tex]
So, the coordinates of the point that divides the line segment internally in the ratio [tex]\(3 : 1\)[/tex] are [tex]\((10.0, 2.0)\)[/tex].

(d) [tex]\( A(2, 6) \)[/tex] and [tex]\( B(12, 1) \)[/tex]; Ratio [tex]\(= 3 : 2\)[/tex]

Using the section formula for internal division:
[tex]\[ x = \frac{3 \cdot 12 + 2 \cdot 2}{3 + 2} = \frac{36 + 4}{5} = 8 \][/tex]
[tex]\[ y = \frac{3 \cdot 1 + 2 \cdot 6}{3 + 2} = \frac{3 + 12}{5} = 3 \][/tex]
So, the coordinates of the point that divides the line segment internally in the ratio [tex]\(3 : 2\)[/tex] are [tex]\((8.0, 3.0)\)[/tex].

(e) [tex]\( A(-7, 5) \)[/tex] and [tex]\( B(3, 0) \)[/tex]; Ratio [tex]\(= 3 : 2\)[/tex]

Using the section formula for internal division:
[tex]\[ x = \frac{3 \cdot 3 + 2 \cdot (-7)}{3 + 2} = \frac{9 - 14}{5} = -1 \][/tex]
[tex]\[ y = \frac{3 \cdot 0 + 2 \cdot 5}{3 + 2} = \frac{0 + 10}{5} = 2 \][/tex]
So, the coordinates of the point that divides the line segment internally in the ratio [tex]\(3 : 2\)[/tex] are [tex]\(( -1.0, 2.0 )\)[/tex].

(f) [tex]\( A(-7, 5) \)[/tex] and [tex]\( B(5, -1) \)[/tex]; Ratio [tex]\(= 5 : 1\)[/tex]

Using the section formula for internal division:
[tex]\[ x = \frac{5 \cdot 5 + 1 \cdot (-7)}{5 + 1} = \frac{25 - 7}{6} = 3 \][/tex]
[tex]\[ y = \frac{5 \cdot (-1) + 1 \cdot 5}{5 + 1} = \frac{-5 + 5}{6} = 0 \][/tex]
So, the coordinates of the point that divides the line segment internally in the ratio [tex]\(5 : 1\)[/tex] are [tex]\((3.0, 0.0)\)[/tex].

In summary, the coordinates are:
[tex]\[ [(2.0, 2.0), (6.0, 4.0), (10.0, 2.0), (8.0, 3.0), (-1.0, 2.0), (3.0, 0.0)] \][/tex]

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