Given:
AB
≅
BC
and AD = 10 in
m∠BDC = 90°
m∠ABC = 130°
Find: m∠ DBC, AC

A) and  

B) and  

Step-by-step explanation:

The picture of the question in the attached figure

Problem A

we know that

Triangle ABC has two equal sides

AB  ≅

BC

so

Is an isosceles triangle

therefore

Remember that

The altitude to the base of an isosceles triangle bisects the vertex angle. and bisects the base

In this problem segment BD represent the altitude to the base of an isosceles triangle ABC

so

Part 1) Find the measure of angle DBC

we have

substitute

Part 2) Find the value of segment AC

we have

substitute

Problem B

we know that

Triangle ABC has two equal sides

AB  ≅

BC

so

Is an isosceles triangle

therefore

Remember that

The altitude to the base of an isosceles triangle bisects the vertex angle. and bisects the base

In this problem segment BD represent the altitude to the base of an isosceles triangle ABC

so

Part 1) Find the measure of angle DBC

we have

substitute

Part 2) Find the value of segment AC

we have

substitute

x =-.20752

step-by-step explanation:

4^(x+2)=12

4^(x+2)=12

take the log base 4 on each side

log 4 (4^(x+2) = log 4(12)

we know log b(a^y) = y log b (a)

(x+2) log 4(4) = log 4 (12)

x+2 = log 4 (12)

we know   logb c = loga c/loga b

we want to convert to base 10   so a = 10

by default, we do not write the 10   c = 4   and b = 10

log4 10 = log 12/log 4

log4 10 = log 12/ log 4

x+2 = log 12/ log 4

subtract 2 from each side

x +2 -2 = log 12/ log 4 -2

x = 1.79248125 -2

x =-.20752