Answer:
5.562148865... cm²
Step-by-step explanation:
we are asked to calculate the area of all 4 shaded (darker) triangles.
it is clear that all 4 triangles are the same size and have therefore the same area.
so, we need to calculate one area and then multiply by 4.
the area of a triangle is
baseline × height / 2
and once we have the baseline, we can calculate the height by using Pythagoras :
2² = height² + (baseline/2)²
so, let's find the baseline :
the white and the dark triangles are similar triangles. that means they have the same angles, and the ratio of the corresponding pairs of sides fit both triangles is the same for all 3 pairs.
so, we see the dark legs are 2cm compared to the large, white triangles of 4cm.
so, the ratio dark triangle / white triangle = 1/2
and therefore the dark baseline is 3×1/2 = 1.5cm
so,
2² = height² + (1.5/2)² = height² + (0.75)² = height² +(3/4)²
4 = height² + 9/16 = height² + 0.5625
height² = 4 - 9/16 = 64/16 - 9/16 = 55/16 = 3.4375
height = sqrt(3.4375) = 1.854049622...cm
the area of such a dark triangle is therefore
1.5 × height / 2 = 0.75 × height = 1.390537216...cm²
4 times this area (and therefore the total area of shaded triangles) is : 5.562148865...cm²
