Question

In the given figure, D is the mid-point of side BC and AE ⊥ BC. If BC = a, AC = b, AB = c, ED = x, AD = p and AE = h, prove that :



(i) $b^2=p^2+ax+\frac{a^2}{4}$
(ii) $c^2=p^2-ax+\frac{a^2}{4}$
(iii) $b^2+c^2=2p^2+\frac{a^2}{2}$

Answer 1

(i) It is given that D is the midpoint of BC and.

Therefore, …......(1)

Using Pythagoras theorem in the right angled triangle AED,

…......(2)

Let us substitute , and in equation (2), we get

Let us take another right angled triangle that is triangle AEC.

Using Pythagoras theorem,

…......(3)

Let us substitute and in equation (3) we get,

Here we know that and .

Substituting , and we get

…......(4)

From equation (1) we can substitute in equation (4),

…......(5)

(ii) Using Pythagoras theorem in right angled triangle AEB,

…...... (6)

We know that AB = c and AE = h now we will find BE.

Therefore,

We know that and substituting these values in we get,

Now we will substitute AB = c, AE = h and in equation (6) we get,

…......(7)

Let us rewrite the equation (7) as below,

…......(8)

From equation (1) we can substitute in equation (8),

…......(9)

(iii) Now we will add equations (5) and (9) as shown below,

Therefore, .