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Distance Formulae

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<h2><span style="font-family:Times New Roman,Times,serif"><span style="font-size:14pt"><span style="color:#0070c0"><strong>Distance Formulae</strong></span></span></span></h2> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">In this lesson, we will study the topic of &nbsp;distance formula in co-ordinate geometry. W will discuss &nbsp;the topic in a comprehensive manner, so that its implementation and application can be understood easily.&nbsp; In mathematics, everything is defined by a formula or a numeric expression. Everything that comes into use or in question is measured in terms of either an equation or by devising out a formula.</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">If we try to understand, distance is defined as space or the length between two points in a given plane. The plane can be two dimensional or three dimensional as well. So, if the two points between which the distance is to be calculated are on the same horizontal plane, the distance between the two can be found by subtracting the co-ordinates that are not the same. However, in terms of analytical geometry, this distance formula is often used to find the distance measure between two lines like the perimeter of polygons on a definite plane, the area of a polygon on a two-dimensional co-ordinate, and many more.</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">For example, we can use the distance formula to calculate the sides of a triangle or to determine whether it&rsquo;s a scalene, isosceles or an equilateral triangle.</span></span></span></p> <h2><span style="font-family:Times New Roman,Times,serif"><span style="font-size:14pt"><span style="color:#0070c0"><strong>Distance Formula Between Two Points</strong></span></span></span></h2> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">Let us now, try and understand this concept as to how to calculate the distance formula between two points in a given plane. In the co-ordinate plane, the distance formula is used to find the distance between any two points. In order to find the distance between two points in the XY plane, we will have to use the distance formula.&nbsp; Here the axis to be assumed will be (X, Y) where X co-ordinate is one and the Y co-ordinate is to be assumed as another. So, we will be able to calculate the distance between these co-ordinates</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">For Example: </span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">Let us consider the following situation</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">A boy walked towards north for 30 meters and then he took a turn towards the east and walked another 40 Meters.&nbsp; we have to calculate the shortest distance from the starting point to the final point &nbsp;</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px"><img alt="distance" src="https://d10lpgp6xz60nq.cloudfront.net/engagement_framework/805502E9-E33E-6665-CEFD-CAD378D2C4B6.webp" style="height:170px; width:280px" />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">&nbsp;Here we have tried to derive it in the pictorial representation for better understanding. Let here the starting point be A and final point be C. the distance between A to B is 30 m and B to C is 40 m.</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">&nbsp;So, the shortest distance between A and C is AC. &nbsp;</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">AC<sup>2</sup>&nbsp;=&nbsp;AB<sup>2</sup>&nbsp;+ BC<sup>2</sup></span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">&nbsp;So, we can calculate it with the square root, </span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">&nbsp;AC-= 50 Ms.</span></span></span></p> <h2><span style="font-family:Times New Roman,Times,serif"><span style="font-size:14pt"><span style="color:#0070c0"><strong><strong>Formula to find distance between two points in 2d plane:</strong></strong></span></span></span></h2> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px"><strong>We try and understand the distance calculation between two points in a 2D plane. Here we will try to calculate the distance between two points in a plane and also devise out the formula for the same.</strong></span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px"><img alt="" src="https://d10lpgp6xz60nq.cloudfront.net/engagement_framework/DCA4D11C-0056-D14E-6828-765151C591C1.webp" style="height:170px; width:280px" /></span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">So, in this diagram, let us consider the points as A (x<sub>1</sub>, y<sub>1)</sub> and B (x<sub>2</sub>, y<sub>2)</sub> in the given co-ordinate. So, in order to calculate the distance between these two points we have the formula as </span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px"><img alt="distance" src="https://d10lpgp6xz60nq.cloudfront.net/engagement_framework/29E3EBB7-556B-2D3F-01EF-713A146DB71F.webp" style="height:70px; width:280px" />&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px"><strong>Related Videos:</strong></span></span></span></p> <ul> <li><span style="font-family:Times New Roman,Times,serif"><span style="font-size:16px"><span style="color:black"><a href="https://doubtnut.com/question-answer-physics/average-velocity-and-average-speed-9773960" style="color:blue; text-decoration:underline">Average Velocity and Average Speed</a></span></span></span></li> <li><span style="font-family:Times New Roman,Times,serif"><span style="font-size:16px"><span style="color:black"><a href="https://doubtnut.com/question-answer-physics/path-or-trajectory-9773956" style="color:blue; text-decoration:underline">Path or Trajectory</a></span></span></span></li> <li><span style="font-family:Times New Roman,Times,serif"><span style="font-size:16px"><span style="color:black"><a href="https://doubtnut.com/question-answer-physics/trajectory-equation-9773957" style="color:blue; text-decoration:underline">Trajectory Equation</a></span></span></span></li> <li><span style="font-family:Times New Roman,Times,serif"><span style="font-size:16px"><span style="color:black"><a href="https://doubtnut.com/question-answer-physics/translatory-motion-9773955" style="color:blue; text-decoration:underline">Translatory Motion</a></span></span></span></li> </ul> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">Let us now take a few solved examples to understand the concept better and its application as well. We will be able to apply in different segments when we are taking the example for the same </span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px"><strong>Example 1:&nbsp;</strong>Find the distance between P (3, -4) and Q (-4, -1).</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px"><strong>Solution: </strong>to find the solution into this </span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">&nbsp;<img alt="distance" src="https://d10lpgp6xz60nq.cloudfront.net/engagement_framework/7D338C96-D2E5-C68F-F73B-F6D36500D91D.webp" style="height:30px; width:200px" /></span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">= &radic; (-4) -(3) + (-1) -(-4))</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">= &radic; (49 + 9)</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">= &radic;58 </span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px"><strong>Example 2: Find the value of a, if the distance between the points P (3, -6) and Q (-3, a) is 10 units.</strong></span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px"><strong>Solution:</strong> By using the distance formula, now we have </span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">Distance between points P (3, -6) and Q (-3, a)</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">&nbsp;=&nbsp;&radic;[(-3-3)<sup>2</sup>+(a+6)<sup>2</sup>]&nbsp;= 10 units</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">So, by Squaring on both sides of the equation will give us,</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">(-6)<sup>2&nbsp;</sup>+ (a+3)<sup>2</sup>&nbsp;=&nbsp;100</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">(a+6)<sup>2</sup>&nbsp;=&nbsp;100 &ndash; 16&nbsp;=&nbsp;64&nbsp;</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">Taking root on both the sides, we get;</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">a + 6&nbsp;=&nbsp;&plusmn;</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px"><strong>In Case I, let us consider a as&nbsp;&nbsp; +8,</strong></span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">a+6= 8,</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">a = 8-6 = 2</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px"><strong>In Case II, let us consider A as -8</strong></span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">a+6&nbsp;= -8</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">a = -8 &ndash; 6</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">a = -14</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">Therefore, the coordinates are either P (3, -6) and Q (-3,2) or P (3, -6) and Q (-3, -14).</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">So, by the Pythagoreans theorem also we can devise out the distance between two points. In mathematics, different concepts are used so as to implement the formula for finding out the distance between the points Let us see another diagram example for the same to make this more clear.</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px"><strong>Example 3 :&nbsp;</strong></span></span></span><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">Here we have to find the distance between the points <em>P</em> (2, 3) and&nbsp;<em>Q</em> (1, 1). O, first of all we have to sketch a diagram to understand the problem in detail. </span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">So according to the hypothesis, we sketch the right-angled triangle&nbsp;<em>PQR</em>&nbsp;with&nbsp;<em>PQ</em>&nbsp;as the hypotenuse.</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px"><img alt="distance" src="https://d10lpgp6xz60nq.cloudfront.net/engagement_framework/525D83D8-E211-B78A-51C8-EF4ED4F6F196.webp" style="height:170px; width:280px" /></span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px"><strong>Solution:</strong><br /> Using Pythagorean theorem:&nbsp;<em>PQ</em><sup>2</sup>&nbsp;= (2 &ndash; 1)<sup>2</sup>&nbsp;+ (3 &ndash;1)<sup>2</sup></span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">&nbsp; &nbsp;&rArr;&nbsp;<em>PQ&nbsp;</em>=&nbsp;<img alt="distance" src="https://d10lpgp6xz60nq.cloudfront.net/engagement_framework/F0A40FE3-0B58-9C93-1026-CB11B647E732.webp" style="height:22px; width:119px" /></span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px"><strong>Example 4:&nbsp;</strong></span></span></span><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">In this example, let&rsquo;s try to Find the distance between the points in a given plane as, A (1, 2) and &nbsp;&nbsp;<em>B</em> (-3, -2).</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px"><strong>Solution:&nbsp;</strong>Using the distance formula:</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">Distance =&nbsp;<img alt="distance" src="https://d10lpgp6xz60nq.cloudfront.net/engagement_framework/BE69F389-5BD8-82DA-3564-DE9C8EC96639.webp" style="height:51px; width:180px" />&nbsp;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">= 5.66 (correct to 2 decimal places)</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">With the help of these examples, we will be in a better place to understand and implement the distance formula.&nbsp; The distance formula is most commonly used in mathematical calculations. This formula is changed a bit when we deal with the different surfaces as in 2 dimensional or three-dimensional plane. </span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px">&nbsp;The distance between a point and another point is calculated using these formulas, and the application of the same in an appropriate manner is very important so that we can do the static calculations for the same.</span></span></span></p> <p><span style="font-family:Times New Roman,Times,serif"><span style="color:#000000"><span style="font-size:16px"><strong>Important Chapters:</strong></span></span></span></p> <ul> <li><span style="font-family:Times New Roman,Times,serif"><span style="font-size:16px"><span style="color:black"><a href="https://doubtnut.com/physics-ncert-solutions" style="color:blue; text-decoration:underline">NCERT Solution for Physics</a></span></span></span></li> <li><span style="font-family:Times New Roman,Times,serif"><span style="font-size:16px"><span style="color:black"><a href="https://doubtnut.com/physics-ncert-solutions/class-11-physics-chapter-3-motion-in-a-straight-line-exercise-1" style="color:blue; text-decoration:underline">NCERT Solution for Class 11 Physics Chapter Motion In A Straight Line</a></span></span></span></li> </ul>

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