The length of [tex]\overline{AC}[/tex] is given by the relationship between the similar triangles
ΔABD and ΔBDC.
Reasons:
The given parameters are;
The altitude of triangle ΔABD = [tex]\overline {BD}[/tex]
The hypotenuse of formed right triangle = [tex]\overline {AC}[/tex]
The length of AD = 8
Length of BD = 24
Whereby ΔABD is a right triangle
We have;
ΔABD is similar to ΔBDC
Therefore, by similar triangle proportional sides relationship, we have;
[tex]\displaystyle \frac{\overline{AD}}{\overline{BD}} = \mathbf{ \frac{\overline{BD}}{\overline{DC}}}[/tex]
Which gives;
[tex]\overline{BD}^2[/tex] = [tex]\mathbf{\overline{DC} \times \overline{AD}}[/tex]
Therefore;
[tex]\displaystyle \overline{DC} = \mathbf{\frac{\overline{BD}^2}{\overline {AD}}}[/tex]
[tex]\displaystyle \overline{DC} = \frac{24^2}{8} = \mathbf{72}[/tex]
[tex]\overline{AC} = \overline{DC} + \overline{AD}[/tex]
Which gives;
[tex]\overline{AC} = 72 + 8 =\mathbf{ 80}[/tex]
Learn more about similar triangles here:
https://brainly.com/question/4618367
