Given right triangle A, B, CABC with altitude start overline, B, D, end overline
BD
drawn to hypotenuse start overline, A, C, end overline
AC
. If A, D, equals, 7AD=7 and A, C, equals, 13, commaAC=13, what is the length of start overline, A, B, end overline
AB
in simplest radical form?
To find the length of AB, we can use the Pythagorean theorem.
Assuming AC is the hypotenuse, we can set up the equation:
AD^2 + BD^2 = AC^2
7^2 + BD^2 = 13^2
49 + BD^2 = 169
BD^2 = 120
BD = √120
BD = 2√30
Now, since BD is the altitude to the hypotenuse, it divides the hypotenuse AC into two segments in the ratio of the adjacent sides of the triangle. Therefore:
AB/BD = AD/DB = 7/2
AB/2√30 = 7/2
AB = 7√30
So, the length of AB in simplest radical form is 7√30.