Question

Group Activity Distance From a Point to a Line This activity investigates how to find the distance from a point \(P(a, b)\) to a line \(L : A x+B y=C\) . (a) Write an equation for the line \(M\) through \(P\) perpendicular to \(L\) (b) Find the coordinates of the point \(Q\) in which \(M\) and \(L\) intersect. (c) Find the distance from \(P\) to \(Q\)

Short answer

The distance from point \(P(a, b)\) to line \(L : A x+B y=C\) is obtained by first getting the equation of line \(M\) perpendicular to \(L\) and passing through \(P\), then finding the point \(Q\) of intersection of \(M\) and \(L\), and finally using the distance formula to calculate the distance from \(P\) to \(Q\).

Step-by-step solution

Find the slope of the line and write the equation

Step 1: Calculate the slope of line \(L\). The equation of line \(L\) is in the standard form \(Ax + By = C\), so the slope is \(-A/B\). The slope of line \(M\), which is perpendicular to \(L\), is the negative inverse of the slope of \(L\), therefore, the slope of line \(M\) is \(B/A\). Now, using the point-slope form of a line, we can write the equation of line \(M\) as \(y - b = B/A * (x - a)\).

Calculate the intersection point of the lines

Step 2: Find the intersection point \(Q\) of lines \(L\) and \(M\). This is done by setting the equations of the two lines equal to each other and solving for \(x\) and \(y\). This will result in a system of equations which can be solved using either substitution or elimination method.

Calculate the distance between points

Step 3: Use the distance formula to find the distance from \(P\) to \(Q\). The distance formula is \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\), where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of point \(P\) and \(Q\) respectively.

Key concepts

Point-Slope Form of a Line

Understanding the point-slope form of a line is essential when attempting to write the equation of a line that is passing through a certain point and has a specific slope. This form is expressed as \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \((x_1, y_1)\) are the coordinates of the point through which the line passes. In this scenario, knowing the slope and one point on the line M, we apply this formula to find its equation. Our line M is found by identifying the perpendicular slope to line L, which is \( B/A \), and using point \( P \'s (a, b)\) coordinates for \( (x_1, y_1) \). Hence, the equation of line M becomes \( y - b = \frac{B}{A}(x - a) \).

When dealing with homework or exam questions, it's significant to carefully substitute the slope and the point's coordinates into the point-slope formula. This careful substitution ensures an accurate equation, essential for the subsequent steps in finding the desired distance.

Perpendicular Lines

Perpendicular lines intersect at a right angle, creating 90-degree corners. They have a special relationship between their slopes: the slope of one is the negative reciprocal of the other. That means if one line has a slope of \( m \), the perpendicular line's slope would be \( -1/m \). This concept is pivotal when determining the equation of the perpendicular line M in our exercise.

For line L with slope \( -A/B \), the slope of its perpendicular line M becomes \( B/A \). Understanding this relationship helps us visualize and correctly ascertain the orientation of lines in geometry. It ensures that line M is indeed perpendicular to L, a crucial step in finding the exact point Q where they intersect.

System of Linear Equations

A system of linear equations consists of two or more equations that share two or more unknowns, usually \( x \) and \( y \). To find a solution to this system—meaning, the values of \( x \) and \( y \) that satisfy all equations simultaneously—you can use methods such as substitution, elimination, or matrix operations. In our example, to find point Q at the intersection of line L and the perpendicular line M, we set their equations equal to each other and solve for \( x \) and \( y \).

This can be a complex process and requires a systematic approach to simplify and solve the equations without making errors. These systems are fundamental in algebra and appear in many real-world applications such as engineering design, economics, and computer science. Employing these methods will lead to the coordinates of point Q, essential in calculating the final distance from point P.

Distance Formula

The distance formula is crucial for calculating the straight-line distance between two points in a coordinate plane. Derived from the Pythagorean theorem, it is given as \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) represent the coordinates of the two points. In the context of our exercise, after establishing the coordinates for point P and the intersection point Q, we use the distance formula to find the length of the line segment connecting them.

This formula is one of the pillars of coordinate geometry and proves its usefulness in various scenarios, from plotting routes in navigation to designing graphics in video games. A clear grasp of the distance formula enables students to address a wide array of geometric problems, culminating in finding the shortest path or distance between points.