Let's solve the problem step by step.
1. Given Information:
- [tex]\( AD = 20 \)[/tex]
- [tex]\( AC = 3x + 4 \)[/tex]
2. Finding [tex]\( x \)[/tex]:
- Since [tex]\( AD = AC + DC \)[/tex] and [tex]\( DC \)[/tex] must be such that the sum equals 20, we can establish the equation where [tex]\( AD = AC \)[/tex] because we need to solve for [tex]\( AC \)[/tex] first.
- Hence, set [tex]\( AD = AC \)[/tex]:
[tex]\[
20 = 3x + 4
\][/tex]
- Solve for [tex]\( x \)[/tex]:
[tex]\[
20 - 4 = 3x
\][/tex]
[tex]\[
16 = 3x
\][/tex]
[tex]\[
x = \frac{16}{3}
\][/tex]
3. Calculating [tex]\( AC \)[/tex]:
- Substitute [tex]\( x = \frac{16}{3} \)[/tex] back into the expression for [tex]\( AC \)[/tex]:
[tex]\[
AC = 3 \left( \frac{16}{3} \right) + 4
\][/tex]
[tex]\[
AC = 16 + 4
\][/tex]
[tex]\[
AC = 20
\][/tex]
4. Calculating [tex]\( DC \)[/tex]:
- Since [tex]\( AD = AC + DC \)[/tex] and we have found that [tex]\( AC = 20 \)[/tex], then:
[tex]\[
20 = 20 + DC
\][/tex]
[tex]\[
DC = 0
\][/tex]
Therefore:
- The value of [tex]\( x \)[/tex] is [tex]\( \frac{16}{3} \)[/tex].
- The value of [tex]\( AC \)[/tex] is 20.
- The value of [tex]\( DC \)[/tex] is 0.
