A regular pentagon has edge length of 7. Find the length of AB, the distance btwn two non-adjacent vertices?

Question

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Thanx

Answer 1

The distance between the vertices is the long side of a triangle having two shorter sides of 7. The included angle is equal to one interior angle of a regular pentagon.

The exterior angles total 360 degrees. Each exterior angle equals 360/5 = 72 degrees. So each interior angle = 108 degrees.

The triangle is isoceles, so each of its base angles equals (180 - 108)/2 = 36 degrees.

Now use the law of sines:

7/sin 36 = x/sin 108

x = 7 sin 108/sin 36
x = 11.326

Answer 2

Not that I don't like the preceding solutions, but we have something way more beautiful here: In a regular pentagon, the ratio of the distance between nonadjacent vertices to the length of a side is phi = (sqrt(5)+1)/2 = 1.618..., the Golden Ratio. So your answer is simply 7*phi = 11.326...

The proof is beautiful in its simplicity. Draw the pentagon ABCDE with sides measuring 1 unit. Draw the segments AC and BD which intersect at point X.

We want to know the length of segment BD. Calculating the angles will reveal two things:

(1) Triangles BXC and BCD are similar, isosceles triangles (36° - 36° - 108°).

(2) Triangle DXC is isosceles (36° - 72° - 72°), so DX = CD = 1. Therefore BD = BX + DX = BX+1.


From (1) we get BD/BC = BC/BX (ratios of similar triangles). Hence

BD = 1/BX (because BC=1)

From (2) we can substitute (BD-1) for BX, so this becomes

BD = 1/(BD - 1)

Which by definition is the Golden Ratio - if you subtract one from it and then take the inverse, you get the same number.

Answer 3

Consider the two sides which connect A to B. Let's assume that one of the sides is the bottom of the pentagon. Then it extends exactly 7 units horizontally and 0 units vertically. Now, the second side intersects the first at an angle of 108 degrees. It extends horizontally 7(-cos 108) units and vertically 7(sin 108) units.

The cosine of 108 degrees is 0.309 and the sine is 0.951, so the second side extends horizontally (0.309 * 7) = 2.163 units and vertically (0.951 * 7) = 6.657 units. That means the total horizontal distance between A and B is 7 + 2.163 = 9.163 units, and the total vertical distance is 6.657 units.

Now, you can use the Pythagorean Theorem to find the total distance between the two points. 9.163^2 + 6.657^2 = 83.963 + 44.321 = 128.284. sqrt(128.284) = 11.326. That's the distance.