Line AD is drawn inside triangle ABC to side BC so that the angle CAD is equal to the angle ACD.

Line AD is drawn inside triangle ABC to side BC so that the angle CAD is equal to the angle ACD. the perimeters of triangles ABC and ABD are 37 and 24, respectively.Find the length of AC

Let’s describe the perimeters of the given triangles:

P (ABC) = AB + BC + AC = 37; 2) P (ABD) = AB + BD + AD. But AD = DC, since by the condition <DAC = <DCA, and triangle ACD is isosceles. If you read equalities 1) – 2), we get:

P (ABC) – P (ABD) = AB + BC + AC – (AB + BD + AD) = AB + BC + AC – AB – BD – AD = BC – BD – AD + AC; (3)

But BC – BD = DC = AD; and P (ABC) – P (ABD) = 37 – 24 = 13;

Consider expression (3), in which BC – BD – AD + AC = P (ABC) – P (AVD) = 13; ВС – ВD – АD = ВС – ВD – DC = 0, AC = 13 remains.