Question Video: Understanding the Effect of the Median in a Triangle by Solving Two Linear Equations | Nagwa Question Video: Understanding the Effect of the Median in a Triangle by Solving Two Linear Equations | Nagwa

Question Video: Understanding the Effect of the Median in a Triangle by Solving Two Linear Equations Mathematics • Second Year of Preparatory School

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In β³πΎππ», πΎπ = 2 and ππ = (5π₯ β 7). Find π₯.

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Video Transcript

In triangle πΎππ», πΎπ equals two and ππ equals five π₯ minus seven. Find π₯.

First, we want to look at our diagram and see what we know. We see that the points π½, π, and πΏ divide each side of the triangles in half. And then we have lines from each of the vertices to those points. The distance from a vertex to a midpoint is the median. And that means π»πΏ, ππ½, and πΎπ are all medians of this triangle. And we know the point of concurrency for three medians is the centroid. We can also write down some other information we know, that πΎπ is two and that ππ is five π₯ minus seven. Because point π is the centroid, πΎπ is equal to two-thirds of πΎπ. That is, the distance from the vertex to the centroid is two-thirds of the distance of the median. And then ππ equals one-third of the distance of πΎπ.

And so we can say that πΎπ is equal to two times ππ. Or we can say that ππ is equal to one-half of πΎπ because two-thirds is equal to one-third times two or one-third is equal to two-thirds times one-half. Using the formula ππ is equal to one-half times πΎπ, we plug in the values we know for ππ and πΎπ. And we get five π₯ minus seven equals one-half times two. One-half times two is one, so we have five π₯ minus seven equals one. And we add seven to both sides. Five π₯ equals eight, and π₯ will equal eight divided by five, which is 1.6.

At this point, itβs probably worth plugging π₯ back in to our five π₯ minus seven to make sure that this answer seems reasonable. We know that ππ will be equal to five times π₯ minus seven. Five times 1.6 is eight; eight minus seven equals ππ. And that means ππ is equal to one. It is true that one is half of two and that two is one times two. This confirms the proportionality of our median and confirms that π₯ equals 1.6.

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