Answered

Join IDNLearn.com today and start getting the answers you've been searching for. Ask anything and receive prompt, well-informed answers from our community of knowledgeable experts.

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]

Sagot :

GinnyAnsweridn
### 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]
We appreciate your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. IDNLearn.com is your source for precise answers. Thank you for visiting, and we look forward to helping you again soon.