1. Triangles

In a Triangle ABC, $\displaystyle |_A = 90 degree$ and D is the mid point of AC.The value of BC^2-BD^2 = AD^2.Prove.

2. Originally Posted by devi
In a Triangle ABC, $\displaystyle |_A = 90 degree$ and D is the mid point of AC.The value of BC^2-BD^2 = AD^2.Prove.
If D is the midpoint of AC, then AD = DC....and AC = 2(AD)

In right triangle ABC,

In right triangle ABD,

Eq,(i) minus Eq.(ii),

Oops...it is not proven.

3. Hello, devi!

The problem has strange wording ... and the statement is not true.

In $\displaystyle \Delta ABC,\;\angle A = 90^o$ and D is the midpoint of AC.
The value of: .$\displaystyle BC^2-BD^2 \:= \:AD^2$ .
. . . "the value of..." ?
Prove.
Consider a 3-4-5 right triangle.
Code:
       B*
| *  *
|   *     *    5
3|     *        *
|       *           *
|         *              *
* - - - - - - * - - - - - - - *
A      2      D       2       C

We have: .$\displaystyle BC = 5,\;BD = \sqrt{13},\;AD = 2$

. . But: .$\displaystyle (5)^2 -(\sqrt{13})^2 \;\;{\bf{\color{red}\neq}}\;\;(2)^2$

4. Triangle BCD is not right-angled so you cannot apply Pythagoras Theorem to it.