Four congruent equilateral triangles are placed in a row. A line connects the bottom left vertex of the first triangle to the top vertex of the last triangle. If each equilateral triangle has an area equal to 6, what is the area above the line contained within the triangles, as shaded in yellow?

As usual, watch the video for a solution.

Triangles In A Row

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Answer To Triangles In A Row Puzzle

(Pretty much all posts are transcribed quickly after I make the videos for them–please let me know if there are any typos/errors and I will correct them, thanks).

Connect the top vertices and construct an upside down equilateral triangle to the left of the first triangle. Suppose each equilateral triangle has a side length s.

Corresponding sides of the congruent equilateral triangles are parallel. Thus we have two sets of similar triangles formed by the top line, the lines slanting to the right (or left), and the halfway dividing line.

The lines slanting to down to the left are proportions of 3 equilateral triangles, so they are in proportions of 1, 2/3, and 1/3. Similarly the lines slanting down to the right are in proportions of 4 equilateral triangles and are in proportions of 1, 3/4, 2/4, 1/4.

The three yellow triangles are similar. The largest yellow triangle has two consecutive sides of s and (3/4)s, and the angle between those sides is 60°, so its area is:

(1/2)s(3/4)s(sin 60°)
= (1/2)s(3/4)s(√3)/2
= (3/4)(s2/4)√3

The area of a single equilateral triangle is 6 = s2(√3)/4. Thus the largest triangle has an area equal to:

(3/4)(s2/4)√3
= (3/4)(6)
= 4.5

The middle yellow triangle has sides that are 2/3 the length of the largest, so its area is (2/3)2 the largest yellow triangle’s area:

= (2/3)2(4.5)
= 2

The smallest yellow triangle has sides that are 1/3 the length of the largest, so its area is (1/3)2 the largest yellow triangle’s area:

= (1/3)2(4.5)
= 0.5

The total area in yellow is thus:

4.5 + 2 + 0.5 = 7