Question

One side of the rectangle lies along the line \(4 \mathrm{x}+7 \mathrm{y}+5=0\). Two of its vertices are \((-3,1)\) and \((1,1)\). Then the equations of other side is (a) \(7 x-4 y+25=0\) (b) \(4 x+7 y=11\) (c) \(7 x-4 y-3=0\) (d) All of these

Short answer

The equation of the other side of the rectangle is \(7x - 4y - 3 = 0\).

Step-by-step solution

Find the slope of the given line

We need to find the slope of the line with the equation \(4x+7y+5=0\). To do this, we will first solve for y. As, \[ 7y = -4x - 5 \Rightarrow y = -\frac{4}{7}x - \frac{5}{7} \] The slope of the given line is \(-\frac{4}{7}\).

Find the midpoint of the line segment

We are given the two vertices of the rectangle: \((-3,1)\) and \((1,1)\). We can find the midpoint of the line segment formed by these points using the midpoint formula: \(M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\) Using the vertices \((-3,1)\) and \((1,1)\) , the midpoint M is calculated as: \(M = \left(\frac{-3+1}{2}, \frac{1+1}{2}\right) = (-1, 1)\)

Find the equation of the line parallel to the given line and passing through the midpoint

We know the slope of the given line and the midpoint of the line segment. Now we can find the equation of the line parallel to the given line and passing through the midpoint. The slope of the line parallel to the given line will also be \(-\frac{4}{7}\). Let's use the point-slope form of the equation: \( y- y_1 = m(x - x_1)\) With slope \(m = -\frac{4}{7}\) and point \((-1, 1)\), plug in the values and solve for y: \[ y - 1 = -\frac{4}{7}(x + 1) \Rightarrow 7y - 7 = -4x - 4 \Rightarrow 7y - 4x = 3 \] Thus, the equation of the line parallel to the given line and passing through the midpoint is \(7y - 4x = 3\).

Identify the correct equation

Based on our calculations, the equation of the other side of the rectangle should be \(7y - 4x = 3\). Comparing this with the given options, we find that option (c) is the correct one. The equation of the other side of the rectangle is: \(7x - 4y - 3 = 0\)

Key concepts

Slope of a Line

The slope of a line tells us how steep a line is and the direction it goes. It is a measure of the rate at which one variable changes with respect to another in a linear relationship. To find the slope from a linear equation, we need to arrange it in a certain format, called the slope-intercept form:
\( y = mx + b \). Here, \( m \) represents the slope, and \( b \) is the y-intercept.

• In the equation \( 4x + 7y + 5 = 0 \), rearrange it to solve for \( y \).
• We find: \( y = -\frac{4}{7}x - \frac{5}{7} \).
• The negative fraction \(-\frac{4}{7}\) is our slope.

This means that for every 7 units we move horizontally to the right, we move 4 units down.

Midpoint Formula

The midpoint formula is a handy tool for finding the exact center point between two known points. It effectively splits a line segment into two equal halves.
Given two points \((x_1, y_1)\) and \((x_2, y_2)\), the midpoint \(M\) is found using:
\( M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right) \).

• For points \((-3,1)\) and \((1,1)\), substitute into the formula.
• \( M = \left(\frac{-3+1}{2}, \frac{1+1}{2}\right) \)
• This simplifies to \((-1, 1)\).

The midpoint is useful for determining center points and constructing symmetrical shapes.

Parallel Lines

Parallel lines run alongside each other, never crossing no matter how far they extend. In geometry, two lines are parallel if they have the same slope.

• Given a line with slope \( m \), any line parallel to it will also have slope \( m \).
• From step one, we know the slope \( m = -\frac{4}{7} \).
• Thus, a line parallel must maintain this slope.

To find such a parallel line, you can use any point along it and apply the slope with the point-slope formula, \( y - y_1 = m(x - x_1) \). This keeps the lines equidistant throughout their entirety.

Equation of a Line

Determining the equation of a line is crucial for defining its path on a plane. We use various forms depending on the given data. For a line passing through a specific point \((x_1, y_1)\) with a given slope \( m \), use the point-slope formula:
\( y - y_1 = m(x - x_1) \).

• Identify known values: slope \( m = -\frac{4}{7} \), point \((-1, 1)\).
• Substitute to find: \( y - 1 = -\frac{4}{7}(x + 1) \).
• Simplify: \( 7y - 4x = 3 \).

This gives the final line equation that is parallel and crosses through the midpoint. Such understanding is fundamental in geometry, particularly with rectilinear shapes like rectangles.