Find the sides and area of an isosceles triangle if the heights dropped to its base and side

| June 16, 2021 | education

Find the sides and area of an isosceles triangle if the heights dropped to its base and side are 5 and 6 cm, respectively.

Let’s calculate the area of the triangle using the formula, in two versions:
S = 1/2 * BD * AC; AC = AD + DC = 2DC; (AD = DC); S = 1/2 * 5 * 2DC = 5DC
S = 1/2 * CF * AB = 1/2 * 6 * AB = 3AB
We get that 5DC = 3AB; 3AB = 3BC; 5DC = 3BC.
Based on the Pythagorean theorem, we get:
BC ^ 2 = BD ^ 2 + DC ^ 2 = 25 + DC ^ 2;
Let us bring the previous equality to the square and substitute the obtained BC ^ 2 into it:
25DC ^ 2 = 9BC ^ 2;
25DC ^ 2 = 9 (25+ DC ^ 2); 25DC ^ 2 = 225+ 9DC ^ 2; 25DC ^ 2-9DC ^ 2 = 225; 16DC ^ 2 = 225; DC ^ 2 = 225/16; Find the roots of the equation DC ^ 2 = 225/16: DC = 15/4, AC = 2DC = 2 (15/4) = 30/4 = 7 2/4 = 7 ½.
Substitute DC in the equality 5DC = 3BC, BC = (5DC) / 3 = (5 * 15/4) / 3 = (75/4) / 3 = 75/12 = 25/4 = 6 ¼ = AB.
S = 3 * 25/4 = 75/4 = 18 ¾.
Answer: AB = BC = 6 1/4, AC = 7 1/2. S = 18 ¾.



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