In a right triangle ABC, right angled at B, altitude BD is drawn to the hypotenuse AC of the triangle. If AD = 6 cm, CD = 5 cm, then find the value of $AB^2+ BD^2$ (in cm).

96

According to the concept,

\( {BD }^{2 } \) =  AD x DC

⇒ \( {BD }^{2 } \) =  6 x 5

⇒ \( {BD }^{2 } \) = 30

According to the concept

\( {AB }^{2 } \) =  AD x AC

⇒ \( {AB }^{2 } \) = 6 x (AD + CD)

⇒ \( {AB }^{2 } \) = 6 x (6 + 5)

⇒ \( {AB }^{2 } \) = 6 x 11

⇒ \( {AB }^{2 } \) = 66

Now, \( {AB }^{2 } \) + \( {BD }^{2 } \) = 30 + 66 = 96

Therefore, \( {AB }^{2 } \) + \( {BD }^{2 } \) is 96.