Given equilateral triangle ABC with medians AD, BE, and CF below.If DO = 3cm and DC = 5.2 cm, what is the area of Triangle ABC?The formula for the area of a triangle is: 𝐴=12𝑏ℎ Area of the triangle = ________ cm²
Question
Given equilateral triangle ABC with medians AD, BE, and CF below.If DO = 3cm and DC = 5.2 cm, what is the area of Triangle ABC?The formula for the area of a triangle is: 𝐴=12𝑏ℎ Area of the triangle = ________ cm²
Solution
In an equilateral triangle, the medians divide each other into segments in the ratio 2:1. Therefore, if DO = 3 cm, then OD = 2 * DO = 6 cm.
Since the medians of an equilateral triangle are equal, AD = BE = CF = DC = 5.2 cm.
The height (h) of the triangle can be found using the Pythagorean theorem: h = sqrt(DC^2 - (OD/2)^2) = sqrt(5.2^2 - 3^2) = sqrt(27.04 - 9) = sqrt(18.04) = 4.24 cm.
The base (b) of the triangle is twice the length of OD, so b = 2 * 6 = 12 cm.
Now we can use the formula for the area of a triangle: A = 1/2 * b * h = 1/2 * 12 cm * 4.24 cm = 25.44 cm².
So, the area of triangle ABC is 25.44 cm².
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