Explanation:
a ratio of 5:5:6 tells us first that there are 5+5+6 = 16 units in all the sides lengths of the triangle.
we don't know how large a single unit is yet, but we know, we have 16 of them.
we also know that all 3 sides together is 32 cm.
and that is another form to bring all side lengths together.
so, we know now that
32 cm = 16 ratio units.
therefore, one ratio unit is 32/16 = 2cm.
therefore, the sides are actually :
2×5 = 10 cm
2×5 = 10 cm
2×6 = 12 cm
so, we have an isoceles triangle (the 2 legs are equally long : 10 cm) with a baseline of 12 cm.
that means the height to the baseline splits the triangle in half (into 2 equal right-angled triangles).
and the height of the main triangle we get via Pythagoras from these smaller triangles :
c² = a² + b²
with c being the Hypotenuse (the baseline opposite of the 90° angle), a and b are the legs.
10² = (baseline/2)² + height² = 6² + height²
100 = 36 + height²
64 = height²
height = 8
the area of a triangle is
baseline × height / 2
so, in our case the area of the triangle is
12 × 8 / 2 = 12 × 4 = 48 cm²