###### Advertisements

###### Advertisements

In 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 + a^2/4`

(ii) `c^2 = p^2 - ax + a^2/4`

(iii) `b^2 + c^2 = 2p^2 + a^2/2`

###### Advertisements

#### Solution

We have, D as the mid-point of BC

(i) AC^{2} = AE^{2} + EC^{2}

b^{2} = AE^{2} + (ED + DC)^{2} [By pythagoras theorem]

b^{2} = AD^{2} + DC^{2} + 2DC × ED

`b^2=p^2+(a/2)^2+2(a/2)xxx` [BC = 2CD given]

`rArrb^2=p^2+a^2/4+ax` .........(i)

(ii) In ΔAEB, by pythagoras theorem

AB^{2} = AE^{2} + BE^{2}

⇒ c^{2} = AD^{2} − ED^{2} + (BD − ED)^{2} [By pythagoras theorem]

⇒ c^{2} = p^{2} − ED^{2} + BD^{2} + ED^{2} − 2BD × ED

`rArrc^2=p^2+(a/2)^2-2(a/2)xx x`

`rArrc^2=p^2+a^2/4-ax` .........(ii)

(iii) Add equations (i) and (ii)

`b^2+c^2=p^2+a^2/4+ax + p^2+a^2/4-ax`

`b^2+c^2=2p^2+(2a^2)/4`

`b^2+c^2=2p^2+a^2/2`

#### APPEARS IN

#### RELATED QUESTIONS

Construct a triangle ABC with sides BC = 7 cm, ∠B = 45° and ∠A = 105°. Then construct a triangle whose sides are `3/4` times the corresponding sides of ∆ABC.

The sides of triangle is given below. Determine it is right triangle or not.

a = 7 cm, b = 24 cm and c = 25 cm

The sides of triangle is given below. Determine it is right triangle or not.

a = 9 cm, b = l6 cm and c = 18 cm

Using Pythagoras theorem determine the length of AD in terms of b and c shown in Figure.

In a right ∆ABC right-angled at C, if D is the mid-point of BC, prove that BC^{2} = 4(AD^{2} − AC^{2}).

ΔABC~ΔDEF such that ar(ΔABC) = 64 cm2 and ar(ΔDEF) = `169cm^2`. If BC = 4cm, find EF.

Find the length of each side of a rhombus whose diagonals are 24cm and 10cm long.

A man goes 12m due south and then 35m due west. How far is he from the starting point.

Find the length of each side of a rhombus are 40 cm and 42 cm. find the length of each side of the rhombus.

Find the diagonal of a rectangle whose length is 16 cm and area is 192 sq.cm ?

Find the side and perimeter of a square whose diagonal is `13sqrt2` cm.

From given figure, In ∆ABC, AB ⊥ BC, AB = BC then m∠A = ?

From given figure, In ∆ABC, AB ⊥ BC, AB = BC, AC = `2sqrt(2)` then l (AB) = ?

From given figure, In ∆ABC, AB ⊥ BC, AB = BC, AC = `5sqrt(2)` , then what is the height of ∆ABC?

Find the height of an equilateral triangle having side 4 cm?

Find the altitude of an equilateral triangle of side 8 cm.

In a ΔABC, ∠CAB is an obtuse angle. P is the circumcentre of ∆ABC. Prove that ∠CAB – ∠PBC = 90°.