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Segment AB with endpoints at A(3,2) and B(3,16) is partitioned by point P according to the ratio of 2:5.

Find the coordinate of point P.

A. (3, 4)
B. (3, 6)
C. (3, 12)
D. (3, 14)

Sagot :

To find the coordinates of point P that partitions the segment AB in the ratio of 2:5, we use the section formula. The section formula for the coordinates of point P dividing the line segment joining points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the ratio [tex]\(m:n\)[/tex] is given by:

[tex]\[ \left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right) \][/tex]

Given:
- Point A has coordinates [tex]\( (x_1, y_1) = (3, 2) \)[/tex]
- Point B has coordinates [tex]\( (x_2, y_2) = (3, 16) \)[/tex]
- The ratio [tex]\( m:n \)[/tex] is [tex]\( 2:5 \)[/tex]

First, calculate the x-coordinate of point P:

[tex]\[ Px = \frac{m \cdot x_2 + n \cdot x_1}{m + n} \][/tex]

Substitute the given values:

[tex]\[ Px = \frac{2 \cdot 3 + 5 \cdot 3}{2 + 5} = \frac{6 + 15}{7} = \frac{21}{7} = 3 \][/tex]

Now, calculate the y-coordinate of point P:

[tex]\[ Py = \frac{m \cdot y_2 + n \cdot y_1}{m + n} \][/tex]

Substitute the given values:

[tex]\[ Py = \frac{2 \cdot 16 + 5 \cdot 2}{2 + 5} = \frac{32 + 10}{7} = \frac{42}{7} = 6 \][/tex]

Therefore, the coordinates of point P are [tex]\( (3, 6) \)[/tex].

Among the given choices:
1. (3,4)
2. (3,6)
3. (3,12)
4. (3,14)

The correct answer is: (3,6).