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What is the [tex]\( x \)[/tex]-coordinate of the point that divides the directed line segment from [tex]\( J \)[/tex] to [tex]\( K \)[/tex] into a ratio of 2:5?

[tex]\[
x = \left(\frac{m}{m+n}\right) \left(x_2 - x_1\right) + x_1
\][/tex]

A. [tex]\(-4\)[/tex]
B. [tex]\(-2\)[/tex]
C. 2
D. 4

Sagot :

GinnyAnswer
To find the [tex]\( x \)[/tex]-coordinate of the point that divides the directed line segment from [tex]\( J \)[/tex] to [tex]\( K \)[/tex] into the ratio [tex]\( 2:5 \)[/tex], we'll use the section formula for internal division in coordinate geometry. The formula is given by:

[tex]\[ x = \left( \frac{m}{m+n} \right) (x_2 - x_1) + x_1 \][/tex]

Here:
- [tex]\( m \)[/tex] and [tex]\( n \)[/tex] are the parts of the ratio.
- [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex] are the [tex]\( x \)[/tex]-coordinates of points [tex]\( J \)[/tex] and [tex]\( K \)[/tex] respectively.

Given:
- [tex]\( m = 2 \)[/tex]
- [tex]\( n = 5 \)[/tex]
- [tex]\( x_1 = -4 \)[/tex]
- [tex]\( x_2 = 4 \)[/tex]

Plugging these values into the formula, we get:

[tex]\[ x = \left( \frac{2}{2+5} \right) (4 - (-4)) + (-4) \][/tex]

Let's break it down step by step:

1. First, calculate [tex]\( m + n \)[/tex]:
[tex]\[ m + n = 2 + 5 = 7 \][/tex]

2. Calculate the fraction [tex]\( \frac{m}{m+n} \)[/tex]:
[tex]\[ \frac{m}{m+n} = \frac{2}{7} \][/tex]

3. Find the difference [tex]\( x_2 - x_1 \)[/tex]:
[tex]\[ x_2 - x_1 = 4 - (-4) = 4 + 4 = 8 \][/tex]

4. Multiply the fraction by the difference:
[tex]\[ \left( \frac{2}{7} \right) \times 8 = \frac{16}{7} \][/tex]

5. Finally, add [tex]\( x_1 \)[/tex] to this result to get the [tex]\( x \)[/tex]-coordinate:
[tex]\[ x = \frac{16}{7} + (-4) = \frac{16}{7} - \frac{28}{7} = \frac{16 - 28}{7} = \frac{-12}{7} \][/tex]

So, the [tex]\( x \)[/tex]-coordinate of the point is:
[tex]\[ x \approx -1.7142857142857144 \][/tex]

Therefore, the [tex]\( x \)[/tex]-coordinate of the point that divides the directed line segment from [tex]\( J \)[/tex] to [tex]\( K \)[/tex] into the ratio [tex]\( 2:5 \)[/tex] is approximately [tex]\(-1.7142857142857144\)[/tex].
devriesivonnemason
To find the \( x \)-coordinate of the point that divides the directed line segment from \( (x_1, y_1) \) to \( (x_2, y_2) \) into a ratio of 2:5, we use the section formula.

Let the coordinates be \( A = (x_1, y_1) \) and \( B = (x_2, y_2) \).

The \( x \)-coordinate \( x \) of the point dividing the segment in the ratio 2:5 is given by:
\[ x = \frac{2 \cdot x_2 + 5 \cdot x_1}{2 + 5} \]

Substituting the values:
\[ x = \frac{2 \cdot 7 + 5 \cdot 1}{2 + 5} \]
\[ x = \frac{14 + 5}{7} \]
\[ x = \frac{19}{7} \]

Therefore, the \( x \)-coordinate of the point that divides the segment from \( (1, 2) \) to \( (7, 5) \) into a ratio of 2:5 is \( \frac{19}{7} \).

So, the correct answer is B. \( \frac{19}{7} \).
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