Question

Find the distance of Line B Your answer should be an integer.
the point of the line is (-9,-10) and (-2,5)
line b is =

Answer 1

To find the distance of line B, we first need to determine the equation of the line that passes through the two given points: \((-9, -10)\) and \((-2, 5)\).

1. **Calculate the slope (m)** of the line using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
where \((x_1, y_1) = (-9, -10)\) and \((x_2, y_2) = (-2, 5)\).

Plugging in the values:
\[
m = \frac{5 - (-10)}{-2 - (-9)} = \frac{5 + 10}{-2 + 9} = \frac{15}{7}
\]

2. **Use the point-slope form** of the line equation:
\[
y - y_1 = m(x - x_1)
\]
Using point \((-9, -10)\) and \(m = \frac{15}{7}\):
\[
y - (-10) = \frac{15}{7}(x - (-9))
\]
Simplifying:
\[
y + 10 = \frac{15}{7}(x + 9)
\]
\[
y + 10 = \frac{15}{7}x + \frac{135}{7}
\]
Subtract \(10\) from both sides:
\[
y = \frac{15}{7}x + \frac{135}{7} - 10
\]
Converting \(10\) to a fraction with a denominator of \(7\):
\[
10 = \frac{70}{7}
\]
Thus:
\[
y = \frac{15}{7}x + \frac{135 - 70}{7} = \frac{15}{7}x + \frac{65}{7}
\]

3. **Write the equation in the standard form** \(Ax + By + C = 0\):
\[
15x - 7y + 65 = 0
\]
gives us \(A = 15\), \(B = -7\), and \(C = 65\).

4. **Finding distance** from the origin (0, 0) to the line:
The distance \(d\) from a point \((x_0, y_0)\) to the line \(Ax + By + C = 0\) is given by:
\[
d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}
\]
Substituting \((x_0, y_0) = (0, 0)\):
\[
d = \frac{|15(0) + (-7)(0) + 65|}{\sqrt{15^2 + (-7)^2}}
\]
Calculating:
\[
d = \frac{|65|}{\sqrt{225 + 49}} = \frac{65}{\sqrt{274}}
\]

Now, compute \(\sqrt{274}\):
\(\sqrt{274} \approx 16.52 \) (not exact but sufficient to find distance).

Calculating:
\[
d \approx \frac{65}{16.52} \approx 3.93
\]

The integer part is 3.

**The distance of line B from the origin is approximately:** \(\boxed{4}\)

Please verify the calculation for rounding based on your specific requirements.