Can you find the slick solution to this geometry problem?
I was unable to find the solution to a problem based on this in an online tutoring session. Afterwards I found the solution the hard way and only much later on did this simple method occur to me. I wonder if this is what the student’s teacher was expecting.
The key to the solution is something that you should have learned in high school geometry. The result is an equation that is simple enough that you could possibly solve it in your head.
Given an equilateral triangle of height 5 inscribed in a larger equilateral triangle such that corresponding sides of the two triangles are parallel and the distance between corresponding sides is 1, what is the height of the larger triangle?
Hint:
Intuitively, what can you say the two triangles share in common?
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8 is the height of the larger triangle. With a suitable diagram of the tops of the two triangles, I can show that the larger triangle’s peak extends above the smaller one’s peak by the hypotenuse of a 30–60-90 triangle whose short side is 1 unit (the separation between triangles). So the peak extends 2 units, while the base extends 1 unit (given), making all together an extra 3 units. 5+8 = 3.
I don’t see a more intuitive shortcut at this point.
8 is the correct answer. Now that I think of it, what I was thinking of may not be all that intuitive, but it is kind of neat how it works out.
The center points of the two triangles coincide. The center points correspond to intersections of the medians (because the triangles are equilateral), which means the distance from the center to the base of the triangles is ⅓ of their heights. The distance from the center to the base of the larger triangle is also 1 greater than the its distance to the shorter triangle. That give ⅓ h = ⅓×5 + 1, or h = 8.
Yes, the center point is ⅓ of the way up, so adding 1 unit below is accompanied by adding 2 units above. I didn’t think of that.
There’s something called Viviani’s theorem (“The sum of the distances from any interior point to the sides of an equilateral triangle equals the length of the triangle’s altitude.”) whereby the result follows directly. By choosing the triangles’ centerpoint, as you suggested, you see that each of the 3 distances is increased 1 unit, which makes their sum – equal to the triangle’s altitude – larger by 3 units.
btw I meant to write 3+5=8 earlier, not 5+8=3 lol. I passed 2nd grade, I think.
I did not know about Viviani’s theorem. Thanks so much for that.
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