Bunuel wrote:

Two roads intersect as shown in the figure above. If RS = ST = TU = UR = 5 meters, what is the straight-line distance, in meters, from S to U?

(A) 5√3/2

(B) 5√2

(C) 15/2

(D) 5√3

(E) 10

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This can be done in one minute with some basic geometry; it is good but not necessary to know that diagonals of a rhombus are perpendicular bisectors. (I did not use that fact directly.)

If all four sides of a quadrilateral are equal, it is a rhombus. A square is a rhombus; this rhombus does not have right angles. It is not a square (so, for example, its four angles are

not all equal).

You do need to know, about a rhombus:

a)

opposite angles are equal

b) its diagonals are angle bisectors

c) like any quadrilateral, sum of interior angles is 360 degrees

NOTE: I am using capital letters S, T, U, and R as the names of the angle measures INSIDE the figure -- that is, as the name of the angles of the vertices

1) Find angle measures in degrees of S, T, U, and R

S = 60 (S's vertical angle = 60)

U = 60 (opposite angles of rhombus are equal)

In diagram, shaded green quarter-circles

S + U = 120

(360 - 120) = 240 degrees remaining

Opposite angles T and R are equal, so their measures are:

\(\frac{240}{2} = 120\) degrees each

In diagram, shaded blue quarter-circles

2) Draw the diagonals, note degree measure of all 4 bisected angles (see diagram)

S and U are now composed of two 30-degree angles

T and R and now composed of two 60-degree angles

Intersecting diagonals create 4 triangles

3) Assess the triangles whose side lengths, added, equal the length of

\(SU\)∆ SAT and ∆ UAT have sides lengths that sum to length of

\(SU\)In both triangles, one angle = 30, one angle = 60. The third must be 90. They are both right triangles

Further, they are 30-60-90 triangles with side ratio

\(x: x√3: 2x\)4) Find side length of SA, and multiply by 2. That is the answer

For 30-60-90 ∆ SAT, the corresponding side lengths from above ratio are:

\(2x = 5\)

\(x = \frac{5}{2}\)

\(x√3 = (\frac{5}{2})√3 = SA\)

\(SA = UA\), and \(SA + UA = SU\)

Hence multiply \(SA\) by 2

\((\frac{5}{2})√3 * (2) = 5√3\)

Answer (D)