Let's solve the problem step-by-step.
### Part (a) – Finding Coordinates of [tex]\(A\)[/tex] and [tex]\(B\)[/tex]
First, we need to find the points where the line [tex]\(\frac{x}{3} + \frac{y}{4} = 1\)[/tex] intersects the [tex]\(x\)[/tex]-axis and the [tex]\(y\)[/tex]-axis.
#### Finding the [tex]\(x\)[/tex]-intercept ([tex]\(A\)[/tex]):
To find the [tex]\(x\)[/tex]-intercept, we set [tex]\(y = 0\)[/tex]:
[tex]\[
\frac{x}{3} + \frac{0}{4} = 1 \implies \frac{x}{3} = 1 \implies x = 3
\][/tex]
So, the [tex]\(x\)[/tex]-intercept is [tex]\(A = (3, 0)\)[/tex].
#### Finding the [tex]\(y\)[/tex]-intercept ([tex]\(B\)[/tex]):
To find the [tex]\(y\)[/tex]-intercept, we set [tex]\(x = 0\)[/tex]:
[tex]\[
\frac{0}{3} + \frac{y}{4} = 1 \implies \frac{y}{4} = 1 \implies y = 4
\][/tex]
So, the [tex]\(y\)[/tex]-intercept is [tex]\(B = (0, 4)\)[/tex].
### Part (b) – Finding the Equation of the Perpendicular Line
Next, we need to find the equation of the line that passes through the point [tex]\((5, -7)\)[/tex] and is perpendicular to the line segment [tex]\(AB\)[/tex].
#### Finding the Slope of Line [tex]\(AB\)[/tex]:
The slope of the line segment [tex]\(AB\)[/tex] is calculated as:
[tex]\[
\text{slope}_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 0}{0 - 3} = \frac{4}{-3} = -\frac{4}{3}
\][/tex]
#### Slope of the Perpendicular Line:
The slope of the line perpendicular to a given line is the negative reciprocal of the given line's slope. Therefore, the slope of the line perpendicular to [tex]\(AB\)[/tex] is:
[tex]\[
\text{slope}_{\text{perpendicular}} = -\frac{1}{\text{slope}_{AB}} = -\frac{1}{-4/3} = \frac{3}{4}
\][/tex]
#### Equation of the Perpendicular Line Passing Through [tex]\((5, -7)\)[/tex]:
We use the point-slope form of a line equation, which is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, [tex]\((x_1, y_1) = (5, -7)\)[/tex] and [tex]\(m = \frac{3}{4}\)[/tex]:
[tex]\[
y - (-7) = \frac{3}{4}(x - 5)
\][/tex]
Simplify this equation:
[tex]\[
y + 7 = \frac{3}{4} (x - 5)
\][/tex]
Distribute [tex]\(\frac{3}{4}\)[/tex]:
[tex]\[
y + 7 = \frac{3}{4}x - \frac{15}{4}
\][/tex]
Subtract 7 from both sides to isolate [tex]\(y\)[/tex]. To combine the constants, convert 7 into quarters [tex]\((7 = \frac{28}{4})\)[/tex]:
[tex]\[
y = \frac{3}{4}x - \frac{15}{4} - \frac{28}{4}
\][/tex]
Combine the fractions:
[tex]\[
y = \frac{3}{4}x - \frac{43}{4}
\][/tex]
So, the equation of the line perpendicular to [tex]\(AB\)[/tex] and passing through the point [tex]\((5, -7)\)[/tex] is:
[tex]\[
y = \frac{3}{4}x - \frac{43}{4}
\][/tex]
### Summary:
1. The coordinates of [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are:
[tex]\[
A = (3, 0) \quad \text{and} \quad B = (0, 4)
\][/tex]
2. The equation of the line that passes through [tex]\((5, -7)\)[/tex] and is perpendicular to the line segment [tex]\(AB\)[/tex] is:
[tex]\[
y = \frac{3}{4}x - \frac{43}{4}
\][/tex]
