In the figure below, C is between A and D, and B is the midpoint of AC. If BC = 8 a...

1. Understand the given information: We are told that point C is between points A and D. Point B is the midpoint of the segment AC. We are given the lengths of two segments: $BC = 8$ and $BD = 10$. We need to find the length of the segment AD.

1. Use the midpoint definition: Since B is the midpoint of AC, it means that the length of segment AB is equal to the length of segment BC. So, $AB = BC$.

1. Calculate the length of AB: We are given $BC = 8$. Therefore, $AB = 8$.

1. Determine the relationship between the segments: Since C is between A and D, the length of segment AD is the sum of the lengths of segments AC and CD. Also, the length of segment AC is the sum of the lengths of segments AB and BC. The length of segment BD is given as 10. We can express BD in terms of other segments. Since C is between B and D, $BD = BC + CD$. Alternatively, since B is between A and C, and C is between A and D, we can also see that $AD = AB + BC + CD$. We also know that $AD = AC + CD$. And $AC = AB + BC$.

1. Calculate the length of AC: The length of AC is the sum of AB and BC. $AC = AB + BC$. Since $AB = 8$ and $BC = 8$, $AC = 8 + 8 = 16$.

1. Calculate the length of CD: We are given $BD = 10$ and we know $BC = 8$. Since C is between B and D, we can write $BD = BC + CD$. Substituting the given values, we get $10 = 8 + CD$. Subtracting 8 from both sides, we find $CD = 10 - 8 = 2$.

1. Calculate the length of AD: The length of AD is the sum of the lengths of AC and CD. $AD = AC + CD$. We found $AC = 16$ and $CD = 2$. Therefore, $AD = 16 + 2 = 18$.

Alternatively, we can express AD as $AD = AB + BD$. Since $AB = 8$ and $BD = 10$, $AD = 8 + 10 = 18$. This is a more direct way to find AD once AB is known.