The midpoint perpendicular to side AB of triangle ABC intersects side BC

The midpoint perpendicular to side AB of triangle ABC intersects side BC at point E. Find AC if BC = 24cm and triangle AEC is 30cm perimeter.

1. ЕН – the middle perpendicular to the AB side. EH divides AB into two equal segments AH = BH. Rectangular triangles AHE and BHE are equal as rectangular in two legs: AH = HB and leg EH is common, then the hypotenuse of triangle AHE AE is equal to the hypotenuse of triangle BHE BE: AE = BE.
2. The perimeter of triangle AEC is equal to:
P = AE + EC + AC.
Since AE = BE, then:
BE + EC + AC = 30 cm.
The AC side consists of two segments BE and EC:
BE + EC = BC;
BE + EC = 24 cm.
Substitute this expression into the expression for the perimeter of the AEC triangle:
24 + AC = 30;
AC = 30 – 24;
AC = 6 cm.
Answer: AC = 6 cm.