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If the coordinates of π΄ and π΅ are five, five and negative one, negative four, respectively, find the coordinates of the point πΆ that divides π΄π΅ internally in the ratio two to one.
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So letβs just picture the situation here.
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We have two points, π΄ and π΅, whose relative positions look something like this.
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Weβre looking to find the coordinates of a point πΆ that divides π΄π΅ internally in the ratio two to one.
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So πΆ is a point somewhere along the length of π΄π΅.
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And the length of the line segment π΄πΆ is twice as long as the length of the segment π΅πΆ.
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Another way of phrasing this is that πΆ is two-thirds of the way along π΄π΅.
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As if π΄π΅ is divided into three equal parts, then two of them are on one side of πΆ and one of them is on the other side.
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To answer this question, Iβm going to think about how we get from π΄ to π΅ in terms of how the coordinates change.
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First of all, Iβll consider the horizontal change.
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At π΄, the π₯-coordinate is five.
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And at π΅, itβs negative one.
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So this is a change of negative six.
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We move six units to the left.
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Next, letβs consider the vertical change.
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At π΄, the π¦-coordinate is five.
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And at π΅, itβs negative four.
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So this is a change of negative nine.
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We move nine units down.
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Letβs think about this ratio then, which has three equal parts.
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Each horizontal and vertical move will be one-third of the total horizontal and vertical move.
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Dividing negative six by three, we have negative two.
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And dividing negative nine by three, we have negative three.
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So each part of this ratio is two units to the left and three units down.
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Remember, πΆ divides this line in the ratio two to one.
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So to get from π΄ to πΆ, we actually move two parts of the ratio.
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Therefore, we need to move four units to the left and six units down.
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To find the coordinates of πΆ, we can therefore apply this transformation to the coordinates of π΄.
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If weβre moving four units to the left, we need to subtract four from the π₯-coordinate.
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So we have five minus four.
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And if weβre moving six units down, we need to subtract six from the π¦-coordinate.
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So we have five minus six.
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This gives the coordinates of πΆ, which are one, negative one.