Question

Help on this question please!

Answer 1

Since every figure repr. an equil. triangle, each of the angles shown is 60 deg.
Thus, the height of one of the shaded triangles is s sin 60 deg, or s/sqrt(3).
The base of one such triangle is s/3.

Thus, the area of ONE shaded triangle is A = bh/2, which here is
        (s/3)(s/sqrt(3) )            (s/3)(s)
A = ----------------------- = --------------------
                   2                          2sqrt(3)

Rationalize the denom.:  Mult numerator and denominator both by sqrt(3), and then mult the resulting numerator and den by 3:

sqrt(3)       s^2/3          s^2*sqrt(3)      s^2*sqrt(3)
----------- * ----------- = ----------------- = ----------------
sqrt(3)      2sqrt(3)       3*2*3                   18

Unfortunately, this does not agree with any of the four answer choices.


Let's take a slightly different approach here:

Each (horizontal) base of each shaded triangle has length s/3.  Each has height (s/3)sin(60 deg) = s*sqrt(3)/(3*2) = s*sqrt(3)/6.

                                                                             [s/3]*[s*sqrt(3)]/6  
Then each shaded triangle has area A = bh/2 = ----------------------------
                                                                                          2

and the total shaded area is three times that:

                      s^2 * sqrt(3)        s^2*sqrt(3)
Total area = -------------------- = ------------------    (answer)
                         3*2  * 6                    36