Question

The diagram below shows equilateral triangle ABC sharing a side with square ACDE. The square has side lengths of 4. What is BE? Justify your answer.

Answer 1

So, we know the sides AB = 4, and AE = 4.

The angle EAB is 90+60 = 150.

The law of the cosines helps:

BE^2 = AE^2 + AB^2 - 2*AB*AE*cos(150) = 4^2 + 4^2 - 2*4*4*(-sqrt(3)/2)

BE^2 = 16+16+16*sqrt(3) = 16(2+sqrt(3)),

BE = 4*sqrt( 2 + sqrt(3 ) ) Nice number

BE ~ 7.7274 ... ~ 7.73


In fact, if you take a ruler, one can see that the ratio between AB and BE is ~ 1.95 which s the same number.

Answer: BE = 4*sqrt( 2 + sqrt(3 ) ) ~ 7.73

Answer 2

BE is $7.73$

What is Square?

In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle with two equal-length adjacent sides.

What is equilateral triangle?

In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each other and are each 60°.

According to question, The diagram below shows equilateral triangle ABC sharing a side with square ACDE.

We have to find BE.

Now, AB $= 4$ and AE $= 4.$

The angle EAB is $90+60 = 150$°

Using the law of the cosines, we get

$BE^2 = AE^2 + AB^2 - 2(AB)(AE)cos(150)$

        $=4^2 + 4^2 -$$2(4)(4)$×$\frac{\sqrt{3} }{2}$

⇒$BE=7.73$

Hence, we can conclude that BE is $7.73$

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