Given an isosceles triangle ABC, AB = BC = 30, the bisectors of the angles a

| September 25, 2021 | education

Given an isosceles triangle ABC, AB = BC = 30, the bisectors of the angles a and c intersect at points M, there is still a segment BM equal to 15, find the area of the triangle.

Let’s construct the height AD, which is the bisector and median of the triangle ABC.

In a right-angled triangle ABD, the segment BM is its bisector, then, by its property:

AM / AB = MD / BD.

MD / BD = 15/30 = 1/2.

BD = 2 * MD.

The length of the segment AD = AM + MD = 15 + MD.

In a right-angled triangle ABD, according to the Pythagorean theorem: AB ^ 2 = AD ^ 2 + BD ^ 2.

900 = (15 + MD) ^ 2 + 4 * MD ^ 2.

900 = 225 + 30 * MD + MD ^ 2 + 4 * MD ^ 2.

5 * MD ^ 2 + 30 * MD – 675 = 0.

МD ^ 2 + 6 * МD – 135 = 0.

Let’s solve the quadratic equation.

МD = 9 cm.

Then AD = 15 + 9 = 14 cm.

ВD = 2 * 9 = 18 cm.

ВС = 2 * ВD = 36 cm.

Then Savs = BC * AD / 2 = 24 * 36/2 = 432 cm2.

Answer: The area of the triangle is 432 cm2.



One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.