+586 In the diagram below, we have \(\angle ABC = \angle CAB = \angle DEB=\angle BDE.\) Given that AE=21 and ED=27, find BD and CA.
+128475 Since angle ABC = angle BDE then in triangle ADB, AB = AD = 21 + 27 = 48
And angle ABC = angle BDE
And angle BDE = angle BDE
So by AA congruency, triangle DEB is similar to triangle DBA
So
DE / DB = DB/ AD
27/ DB = DB/ 48
27 * 48 = DB^2
1296 = DB^2
36 = DB
The EB also = 36
And because angle ABC = angle CAB = angle DEB = angle BDE, then by AA congruency, triangle BAC is similar to triangle DEB
So
CA / BA = EB / DE
CA/ 48 = 36 / 27
CA / 48 = 4/3
CA = 48 * 4 / 3
CA = 48/3 * 4
CA = 16 * 4
CA = 64
