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

In △𝐾𝑀𝐻, 𝐾𝑄 = 2 and 𝑄𝑃 = (5π‘₯ βˆ’ 7). Find π‘₯.

02:38

Video Transcript

In triangle 𝐾𝑀𝐻, 𝐾𝑄 equals two and 𝑄𝑃 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 𝐽, 𝑃, and 𝐿 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 𝐻𝐿, 𝑀𝐽, and 𝐾𝑃 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 𝐾𝑄 is two and that 𝑄𝑃 is five π‘₯ minus seven. Because point 𝑄 is the centroid, 𝐾𝑄 is equal to two-thirds of 𝐾𝑃. That is, the distance from the vertex to the centroid is two-thirds of the distance of the median. And then 𝑄𝑃 equals one-third of the distance of 𝐾𝑃.

And so we can say that 𝐾𝑄 is equal to two times 𝑄𝑃. Or we can say that 𝑄𝑃 is equal to one-half of 𝐾𝑄 because two-thirds is equal to one-third times two or one-third is equal to two-thirds times one-half. Using the formula 𝑄𝑃 is equal to one-half times 𝐾𝑄, we plug in the values we know for 𝑄𝑃 and 𝐾𝑄. 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, it’s probably worth plugging π‘₯ back in to our five π‘₯ minus seven to make sure that this answer seems reasonable. We know that 𝑄𝑃 will be equal to five times π‘₯ minus seven. Five times 1.6 is eight; eight minus seven equals 𝑄𝑃. And that means 𝑄𝑃 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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