The height ad of the triangle abc divides the side bc into segments bm and mc.

The height ad of the triangle abc divides the side bc into segments bm and mc. Find the length of the segment mc if ab = 10√2 cm, ac = 26cm, and the angle B = 45 degrees.

The solution of the problem:

Angle bad = 90 ° – 45 ° = 45 ° and is equal to angle b.
Triangle abd is isosceles, side ad = bm.

Determine the height ad.
ab ^ 2 = ad ^ 2 + bm ^ 2;

ab ^ 2 = 2ad ^ 2;

ad ^ 2 = ab ^ 2/2;

ad ^ 2 = 10√2 / 2 = 5√2;

ad = √ (5√2) cm.

Let’s define the segment mc.
mc ^ 2 = ac ^ 2 – ad ^ 2;

mc ^ 2 = 26 ^ 2 – 5√2;

mc ^ 2 = 676 – 5 * 1.4;

mc ^ 2 = 683;

mc = √683;

mc = 26.13 cm.

Answer: the length of the segment mc is 26.13 cm.