Answer: [tex]\overline{AC}=[/tex] 12.5
Step-by-step explanation: The triangle ABC and its altitude [tex]\overline{BD}[/tex] is represented in the figure below.
Altitude is a segment of line that link a vertex and the opposite side, forming a right angle.
So, because of [tex]\overline{BD}[/tex], now we have two similar triangles, which means that ratios of corresponding sides are equal:
[tex]\frac{\overline{BD}}{\overline{BC}} =\frac{\overline{AD}}{\overline{BD}}[/tex]
[tex]\overline{BD}^{2}=\overline{BC}.\overline{AD}[/tex] (1)
This is always true for a right triangle and a altitude drawn to the hypotenuse.
Triangle BDC is also right triangle. So, we can use Pythagorean theorem to determine the missing side.
[tex]\overline{BC}^{2}=\overline{CD}^{2}+\overline{BD}^{2}[/tex]
[tex]\overline{BD}^{2}=\overline{BC}^{2}-\overline{CD}^{2}[/tex] (2)
Substituting (2) into (1):
[tex]\overline{BC}^{2}-\overline{CD}^{2}=\overline{BC}.\overline{AD}[/tex]
We want to find f, so:
[tex]\overline{AD}=\frac{\overline{BC}^{2}-\overline{CD}^{2}}{\overline{BC}}[/tex]
[tex]f=\frac{10^{2}-5^{2}}{10}[/tex]
f = 7.5
The length of [tex]\overline{AC}[/tex] is
[tex]\overline{AC}=f+5[/tex]
[tex]\overline{AC}=7.5+5[/tex]
[tex]\overline{AC}=12.5[/tex]
The length of the hypotenuse of triangle ABC is 12.5 units.
