Question Video: Finding the Coordinates of Points Using the Midpoint Formula | Nagwa Question Video: Finding the Coordinates of Points Using the Midpoint Formula | Nagwa

# Question Video: Finding the Coordinates of Points Using the Midpoint Formula Mathematics • Third Year of Preparatory School

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Consider the points π΄ (7, 7), π΅ (9, β7), and πΆ (5, 1). Given that line segment π΄π· is a median of the triangle π΄π΅πΆ and π is the midpoint of this median, determine the coordinates of π· and π.

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### Video Transcript

Consider the points π΄ seven, seven; π΅ nine, negative seven; and πΆ five, one. Given that line segment π΄π· is a median of the triangle π΄π΅πΆ and π is the midpoint of this median, determine the coordinates of π· and π.

We begin by sketching the triangle π΄π΅πΆ so we can see what weβre looking for. Weβre told in the question that line segment π΄π· is a median of the triangle and that π is the midpoint of this median. And we want to determine the coordinates of π· and π. Now, we know that a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. In our case, the vertex is the point π΄, π· is the midpoint of the opposite side, πΆπ΅, and π is the midpoint of the median.

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. 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. And once weβve found our point π·, we can use this to find the midpoint π of the line segment π΄π· in the same way.

Okay, so π· is the midpoint of the line segment πΆπ΅. And so, our two points π₯ one, π¦ one and π₯ two, π¦ two are πΆ five, one and π΅ nine, negative seven. 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 thatβs the π₯-coordinate and π¦-coordinate one plus negative seven over two. That is 14 over two and negative six over two. So, the π· has coordinates seven, negative three. And we see that this agrees with the position of π· on our sketch.

So, we have our point π·, which is the midpoint of line segment πΆπ΅, and now we want to find the midpoint π of line segment π΄π·. 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. π₯ one plus π₯ two over two is seven plus seven over two. And the π¦-coordinate π¦ one plus π¦ two over two is equal to seven plus negative three over two. 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.

The coordinates of the two points π· and π are therefore π· is seven, negative three and π is seven, two.

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