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Consider the points π΄ seven, seven; π΅ nine, negative seven; and πΆ five, one.
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Given that line segment π΄π· is a median of the triangle π΄π΅πΆ and π is the midpoint of this median, determine the coordinates of π· and π.
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We begin by sketching the triangle π΄π΅πΆ so we can see what weβre looking for.
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Weβre told in the question that line segment π΄π· is a median of the triangle and that π is the midpoint of this median.
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And we want to determine the coordinates of π· and π.
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Now, we know that a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side.
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In our case, the vertex is the point π΄, π· is the midpoint of the opposite side, πΆπ΅, and π is the midpoint of the median.
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Our order of play then is to first find the point π· using the formula for the midpoint of a line segment between two points.
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This tells us that for two points with coordinates π₯ one, π¦ one and π₯ two, π¦ two, the midpoint of the line segment between them has coordinates π₯ one plus π₯ two over two and π¦ one plus π¦ two over two.
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And once weβve found our point π·, we can use this to find the midpoint π of the line segment π΄π· in the same way.
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Okay, so π· is the midpoint of the line segment πΆπ΅.
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And so, our two points π₯ one, π¦ one and π₯ two, π¦ two are πΆ five, one and π΅ nine, negative seven.
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And substituting these into the formula for the midpoint, we have five plus nine over two β that is, π₯ one plus π₯ two over two, and thatβs the π₯-coordinate and π¦-coordinate one plus negative seven over two.
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That is 14 over two and negative six over two.
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So, the π· has coordinates seven, negative three.
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And we see that this agrees with the position of π· on our sketch.
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So, we have our point π·, which is the midpoint of line segment πΆπ΅, and now we want to find the midpoint π of line segment π΄π·.
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So now to use our formula for π, we let π΄ be the point π₯ one, π¦ one with coordinates seven, seven and π· be the point π₯ two, π¦ two with coordinates seven, negative three.
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π₯ one plus π₯ two over two is seven plus seven over two.
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And the π¦-coordinate π¦ one plus π¦ two over two is equal to seven plus negative three over two.
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That is, π₯ is 14 over two and π¦ is four over two so that our midpoint π has coordinates seven, two, which again agrees with the position of π on our sketch.
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The coordinates of the two points π· and π are therefore π· is seven, negative three and π is seven, two.