# In an isosceles triangle ABC AB = CB. Height BD = 1.5 cm, AC: AB = 4: 5. Find the area of a triangle.

September 12, 2021 | education

| Solution:

Since AC: AB = 4: 5, then let AC = 4x, AB = 5x.

In an isosceles triangle, the height drawn to the base is the median, therefore:

AD = AC: 2 = 4x: 2 = 2x.

By the Pythagorean theorem in the triangle ABD:

AB ^ 2 = AD ^ 2 + BD ^ 2;

(5x) ^ 2 = (2x) ^ 2 + 1.5 ^ 2;

25x ^ 2 = 4x ^ 2 + 2.25;

25x ^ 2 – 4x ^ 2 = 2.25;

21x ^ 2 = 2.25;

x ^ 2 = 2.25 / 21;

x ^ 2 = 225/2100;

x ^ 2 = 3/28;

x = √ (3/28);

x = √3 / 2√7;

AC = 4x = 4√3 / 2√7 = 2√3 / √7 = 2√21 / 7.

S (ABC) = 1 / 2AC ∙ BD = 1/2 ∙ 2√21 / 7 ∙ 1.5 = 3√21 / 14.

Answer: S (ABC) = 3√21 / 14.

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