In triangle ABC, AB =10 and AC =17. Let D be the foot of the perpendicular from A to BC. If BD:CD = 2:5, then find AD.
https://web2.0calc.com/questions/in-triangle-abc-ab-10-and-ac-17-let-d-be-the-foot
^an amazing explanation by CPhill^
AB = 10 AC = 17
BD : DC = 2 : 5 BD = 2x DC = 5x
√(AB2 - BD2) = AD ⇒√(100 - 4x2)= AD √(AC2 - DC2) = AD ⇒√( - 25x2) = AD Therefore, √(100 - 4x2) = √(289 - 25x2) ⇒100 - 4x2 = 289 - 25x2 ⇒-189 = -21x2
⇒189 = 21x2
⇒21x2 - 189 = 0 ⇒ 21(x2 - 9) = 0 ⇒x2 - 9 = 0 ⇒ x = 3
Now, AD = √(AB2 - BD2) = √(100 - 4(x)2)
= √(100 - 4(3)2) = √(100 - 36) = √(64) = 8
+76 
