Distance Between Two Parallel Lines

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You can join any two given points in multiple ways; it could be infinite, to say so. But if you are asked to join in a way that gives you the shortest distance between two points, there is only one way to do it. And that is through a straight line. 

A straight line is a one-dimensional figure that joins two points, and it can extend to an infinite length on either side of the points. A straight line has zero curvature, and it can go vertically (up and down) or horizontally (left to right). Some of the properties of a straight line are:

  • They move on a straight path without changing direction.

  • The slope of the line (also called its gradient) is a measure of its vertical change to horizontal change. So on the x-y coordinate axis, the slope (m) is given by m = ∆y/∆x. In other words, it is a measure of the steepness of a line.

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  • Equation of a straight line is given as y = m (gradient of the line) * x + c (intercept of the line on the y axis)

  • A straight line has only length property but no breadth.

  • Straight lines can be parallel, concurrent, intersecting, or perpendicular to each other.

  • We can also express the slope or gradient of a straight line by its angle with the x-axis. If the angle between the x-axis and straight line is θ, then m (slope) = tan θ.

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In the graph above, the equation of the line is y = 4/3 * x - 2, as you can see the line cuts the y-axis at the point -2.

Let us understand some terminologies around straight lines, a distance of a line formula, length of line formula, etc., before delving into other topics like the distance between a point and a line formula and distance between two parallel lines.

  • Line segment - It is a portion of the straight line, which is the distance between two points falling on the line.

  • Parallel lines - We see them in daily lives like railway tracks, ladder rungs, etc. Two lines are parallel when they never meet in space. They maintain the same distance between each other if extended till infinity. All through their paths, these lines are not inclined towards each other at any angle.

Distance Between a Point and a Line Formula

When you need to find the distance between two objects, let us say the distance between two street lamps, you would measure the length between them. 

Now we will see how you can find the distance of the line from the point when the point is not lying on the line. To find the distance between point and line, we first need to know the distance formula, which is:

XY = \[\sqrt{{((x2 - x1)^{2} + y2 - y1)^{2}}}\]

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Let us say there is a line l in the XY-coordinate plane and P is any point at a distance of r from the line. Then:

  • The equation of this line is Xx + Yy + C = 0.

  • The length of the perpendicular drawn from point P on the line l is P's distance from l.

  • The x and y-intercepts are -CX and -CY, respectively.

  • The line meets the x-axis at X and the y-axis at Y, respectively.

  • The coordinates of points X and Y are X (0, -C/Y) and Y (-C/X, 0).

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We calculate the area of the triangle XPY as:

 area (△XPY) = 1/2 * base * height = ½ * XY * PM

 So, PM = 2 *  area (△XPY) /XY ---- (I)

 area (△XPY) is also given by ½ * |x1(y2 − y3) + x2(y3 − y1) + x3(y1 − y2)|

Or, ½ | x1 (0 + C/Y) + (−C/X) (−C/Y −  y1) +0 ( y1 − 0)| = ½ |x1 * C/Y +  y1 * C/X + C2/XY|

Or, ½ |C/ (XY) |.|Xx1 + Yy1 + C|… (II)

Distance of the line XY = ((0 + C/X)2 + (C/Y − 0)2)½ = |C| × ((1/X2) + (1/Y2))½

Or, Distance, XY = |C/XY| (X2 + Y2)½ … (III)

Combining (I), (II), and (III) we can write:

PM = r = |Ax1 + Yy1 + C| / (X2 + Y2)½

 

Distance Between Two Parallel Lines 

Two lines that never intersect, even when extended till the infinity, are parallel. 

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Two parallel lines have equal slopes so:

  • Let there be a line with the equation “ax + by + c = 0” 

  • A line parallel to this above line will be represented by equation “ax + by + t = 0”. 

Now let us know how to find the distance between them.

  • Since the slope of two || lines are the same, they can be written as :

                             y = mx + k1 -> Line 1

                              y = mx + k2 -> Line 2

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  • Line 1 intersects the X-axis at the point P (-k1/m, 0).

  • The distance between lines 1 and 2 is the length of the perpendicular from line 1 on line 2 which is:

|(–m)(–k1/m) + (–k2)|/√(1 + m2) OR

 d = |k1–k2|/√(1+m2).

FAQs (Frequently Asked Questions)

Q1: What are the Properties of the Angles Which are Associated with Parallel Lines?

Ans: 

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If a transversal cuts two parallel lines then:

  • The pair of corresponding angles are equal. So in the diagram shown above:

∠2 = ∠6, ∠1 = ∠5, ∠3 = ∠7, and ∠4 = ∠8

  • The interior alternate angles are equal i.e.

∠4 = ∠6 and ∠3 = ∠5.

  • The exterior alternate angles are equal i.e.

∠1 = ∠7 and ∠2 = ∠8.

  • Interior angles that lie on the same side of the transversal are supplementary i.e. their sum is 180 degrees. Hence:

∠3 + ∠6 = 180°

∠4 + ∠5 = 180°.

Q2: What is Meant by the Collinearity of Three Points?

Ans: If three points lie on the same straight line, then they are called collinear points. Another condition that collinear points satisfy is that if the points are X, Y, and Z (and Y is between X and Z), then the slope of XY = slope of YZ, i.e. (b2 − b1)/(a2 − a1) = (b3 − b2)/(a3 − a2).

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