# 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.