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In this explainer, we will learn how to find the perpendicular bisector of a line segment by identifying its midpoint and finding the perpendicular line passing through that point. Our first objective is to learn how to calculate the coordinates of the midpoint of a line segment connecting two points. Suppose we are given two points π(π₯,π¦)ο§ο§ο§ and π(π₯,π¦)ο¨ο¨ο¨. The midpoint of the line segment ππο§ο¨ is the point π lying on ππο§ο¨ exactly halfway between πο§ and πο¨. This means that the π₯-coordinate π₯ο of π lies halfway between π₯ο§ and π₯ο¨ and may therefore be calculated by averaging the two points, giving us π₯=π₯+π₯2οο§ο¨. The same holds true for the π¦-coordinate π¦ο of π. This leads us to the following formula. Suppose π(π₯,π¦)ο§ο§ο§ and π(π₯,π¦)ο¨ο¨ο¨ are points joined by a line segment ππο§ο¨. Then, the coordinates of the midpoint of the line segment ππο§ο¨ are given by π(π₯,π¦)=οΌπ₯+π₯2,π¦+π¦2ο.οοο§ο¨ο§ο¨ Let us practice finding the coordinates of midpoints. Given π΄(4,8) and π΅(6,6), what are the coordinates of the midpoint of π΄π΅? We recall that the midpoint π of a line segment is the point halfway between the endpoints, which we can find by averaging the π₯- and π¦-coordinates of π΄ and π΅ respectively. Thus, we apply the formula: π(π₯,π¦)=οΌπ₯+π₯2,π¦+π¦2ο=οΌ4+62,8+62ο=(5,7).οοο§ο¨ο§ο¨ Therefore, the coordinates of the midpoint of π΄π΅ are (5,7). We can also use the formula for the coordinates of a midpoint to calculate one of the endpoints of a line segment given its other endpoint and the coordinates of the midpoint. The origin is the midpoint of the straight segment π΄π΅. Find the coordinates of point π΅ if the coordinates of point π΄ are (β6,4). Here, we have been given one endpoint of a line segment and the midpoint π(0,0) and have been asked to find the other endpoint. We can do this by using the midpoint formula in reverse: π(π₯,π¦)=(0,0)=οΌπ₯+π₯2,π¦+π¦2ο=ο½β6+π₯2,4+π¦2ο.οοοΊο»οΊο»ο»ο» This gives us two equations: β6+π₯2=0β6+π₯=0π₯=6ο»ο»ο» and 4+π¦2=04+π¦=0π¦=β4.ο»ο»ο» We conclude that the coordinates of π΅ are (6,β4). One application of calculating the midpoints of line segments is calculating the coordinates of centers of circles given their diameters for the simple reason that the center of a circle is the midpoint of any of its diameters. In the next example, we will see an example of finding the center of a circle with this method. Points π΄(4,1) and π΅(β4,β5) define the diameter π΄π΅ of a circle with center π. Find the coordinates of π and the circumference of the circle, rounding your answer to the nearest tenth. The center π of the circle is the midpoint of its diameter π΄π΅. Recall that the midpoint π of a line segment (such as a diameter) can be found by averaging the π₯- and π¦-coordinates of the endpoints π΄ and π΅ as follows: π(π₯,π¦)=οΌπ₯+π₯2,π¦+π¦2ο=οΌ4β42,1β52ο=(0,β2).οοοΊο»οΊο» The circumference of a circle is given by the formula πΆ=2ππ, where π is the length of its radius. The length of the radius is the distance from the center of the circle to any point on its radius, for example, the point π΄(4,1). We can calculate this length using the formula for the distance between two points π and π΄: π=(π₯βπ₯)+(π¦βπ¦)π=(0β4)+(β2β1)=16+9=25.ο¨οοΊο¨οοΊο¨ο¨ο¨ο¨ Taking the square roots, we find that π=5 and therefore the circumference is 2ππ=10π=31.4 to the nearest tenth. In conclusion, the coordinates of the center are (0,β2) and the circumference is 31.4 to the nearest tenth. We turn now to the second major topic of this explainer, calculating the equation of the perpendicular bisector of a given line segment. Given a line segment π΄π΅, the perpendicular bisector of π΄π΅ is the unique line perpendicular to π΄π΅ passing through the midpoint of π΄π΅. Recall that for any line πΏ with slope π, the slope of any line perpendicular to it is the negative reciprocal of π, that is, β1π. We can use this fact and our understanding of the midpoints of line segments to write down the equation of the perpendicular bisector of any line segment. Suppose we are given a line segment ππο§ο¨ with endpoints π(π₯,π¦)ο§ο§ο§ and π(π₯,π¦)ο¨ο¨ο¨ and want to find the equation of its perpendicular bisector.
Let us have a go at applying this algorithm. Find the equation of the perpendicular bisector of the line segment joining points π΄(1,3) and π΅(7,11). Give your answer in the form π¦=ππ₯+π. AnswerTo find the equation of the perpendicular bisector, we will first need to find its slope, which is the negative reciprocal of the slope of the line segment joining π΄ and π΅. This is given by slopeofπ΄π΅=π¦βπ¦π₯βπ₯=11β37β1=86=43.ο¨ο§ο¨ο§ Now, we can find the negative reciprocal by flipping over the fraction and taking the negative; this gives us the following: slopeofperpendicularbisector=β34. Next, we need the coordinates of a point on the perpendicular bisector. Since the perpendicular bisector (by definition) passes through the midpoint of the line segment π΄π΅, we can use the formula for the coordinates of the midpoint: π(π₯,π¦)=οΌπ₯+π₯2,π¦+π¦2ο=οΌ1+72,3+112ο=(4,7).οοο§ο¨ο§ο¨ Substituting these coordinates and our slope β34 into the pointβslope form of the equation of a straight line, π¦βπ¦=π(π₯βπ₯)π¦β7=β34(π₯β4),ο§ο§ and rearranging into the form π¦=ππ₯+π, we have π¦=β34π₯+10. For our last example, we will use our understanding of midpoints and perpendicular bisectors to calculate some unknown values. A line segment π΄π΅ joins the points π΄(β6,β6) and π΅(0,π). The perpendicular bisector of π΄π΅ has equation π¦=β3π₯+π. Find the values of π and π. AnswerSince the perpendicular bisector has slope β3, we know that the line segment π΄π΅ has slope 13 (the negative reciprocal of β3). We can calculate the π¦-coordinate of point π΅ (that is, π) by using the definition of the slope π: π=π¦βπ¦π₯βπ₯13=πβ(β6)0β(β6)=π+662=π+6π=β4.ο»οΊο»οΊ We will calculate the value of π in the equation π¦=β3π₯+π of the perpendicular bisector using the coordinates of the midpoint π of π΄π΅ (which is a point that lies on the perpendicular bisector by definition). We have the formula π(π₯,π¦)=οΌπ₯+π₯2,π¦+π¦2ο=οΌβ6+02,β6β42ο=(β3,β5).οοο§ο¨ο§ο¨ We can now substitute π₯=β3 and π¦=β5 into the equation of the perpendicular bisector and rearrange to find π: π¦=β3π₯+πβ5=β3Γβ3+ππ=β5β9=β14. Our solution to the example is π=β4, π=β14. Let us finish by recapping a few important concepts from this explainer.
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