Part A:
BD being perpendicular to AC allows you to know that you are dealing with RIGHT triangles!
The fact that you are dealing with right Triangles allows you to use the Pythagorean theorem.
B:
Line segment AD can be found by using the pythagorean theorem.
if a^2 + b^2 = c^2 (it does), then we can say AD is a, BD is b, and AB is c.
a^2 + (8)^2 = (10)^2
a^2 + 64 = 100
a^2 = 100 - 64 = 36
a = root(36) = 6
so AD = 6
C:
You may want to elaborate more than this, but you can see that AB and BC are equal, meaning this is an isosceles triangle. So, logically, the perpendicular we drew is exactly half-way between AC. So AD = DC. 6 = DC. AD + DC = AC. 6 + 6 = AC = 12
D (though it says C):
The perimeter of ABD is just AB + BC + AC. AB is 10, BC is 10 and AC is 12
10 + 10 + 12 = 32
I hope that helped.