Given:
A triangle ABC where AD = 12 units, AB = 15 units, and AC = 13 units.
To find:
The length of BC.
Solution:
According to the Pythagorean theorem, the square of the hypotenuse will be equal to the sum of the squares of the other two sides.
There are 2 triangles, ACD and ABD in triangle ABC.
In triangle ACD, AC is the hypotenuse.
In triangle ABD, AB is the hypotenuse.
Assume the length of CD is x and the length of DB is y.
For triangle ACD, [tex]AC^{2} = AD^{2} + CD^{2} .[/tex]
[tex]13^{2} = 12^{2} +x^{2} .[/tex]
[tex]x^{2} = 169-144 = 25.[/tex]
[tex]x = \sqrt{25} = 5.[/tex]
For triangle ABD, [tex]AB^{2} = AD^{2} + BD^{2} .[/tex]
[tex]15^{2} = 12^{2} +y^{2} .[/tex]
[tex]y^{2} = 225-144 = 81.[/tex]
[tex]y = \sqrt{81} = 9.[/tex]
[tex]CB=CD+DB = 5+9=14[/tex] units.
So BC is option C. 14 units long.
