In an isosceles triangle ABC AB = BC = 40 cm, AC = 20 cm, point H is marked on the BC side so that BH: HC = 3: 1. find AH.

Consider a triangle ΔАHС.

In order to find the length of the segment AH, we use the formula for the length through two sides and the angle between them:

a = √b ^ 2 + c ^ 2 – 2 b c cos α.

To do this, we need to find the length of the segment C and ∠C.

Since the ratio BH / HC = 3/1, then

HC = x;

BH = 3x;

BC = 40;

x + 3x = 40;

4x = 40;

x = 40/4 = 10 cm.

HC = 10 cm;

BH = 10 3 = 30 cm.

In order to find cos C in the triangle ΔABC, we draw the height BK from angle B. According to the cosine theorem, cosine is the ratio of the adjacent leg to the hypotenuse:

cos С = КС / ВС;

Since the triangle is isosceles, the KC segment is equal to half of the AC segment:

KC = AC / 2;

KС = 20/2 = 10 cm;

cos C = 10/40 = 0.25.

Now we find the length of the segment АН in the triangle ΔАНС:

AH = √HC ^ 2 + AC ^ 2 – 2 * HC * * AC * cos C;

AH = √10 ^ 2 + 20 ^ 2 – 2 ∙ * 10 * ∙ 20 ∙ * 0.25 = √100 + 400 – 100 = √400 = 20 cm.

Otvnt: the length of the segment AH is 20 cm.