###### Advertisements

###### Advertisements

In right-angled triangle ABC in which ∠C = 90°, if D is the mid-point of BC, prove that AB^{2} = 4AD^{2} − 3AC^{2}.

###### Advertisements

#### Solution

We have,

∠C = 90° and D is the mid-point of BC

In ΔACB, by Pythagoras theorem

AB^{2} = AC^{2} + BC^{2}

⇒ AB^{2} = AC^{2} + (2CD)^{2} [D is the mid-point of BC]

AB^{2} = AC^{2} + 4CD^{2}

⇒ AB^{2} = AC^{2} + 4(AD^{2} − AC^{2}) [In ΔACD, by Pythagoras theorem]

⇒ AB^{2} = AC^{2} + 4AD^{2} − 4AC^{2}

⇒ AB^{2} = 4AD^{2} − 3AC^{2}

#### APPEARS IN

#### RELATED QUESTIONS

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

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

A man goes 15 metres due west and then 8 metres due north. How far is he from the starting point?

A ladder 17 m long reaches a window of a building 15 m above the ground. Find the distance of the foot of the ladder from the building.

A triangle has sides 5 cm, 12 cm and 13 cm. Find the length to one decimal place, of the perpendicular from the opposite vertex to the side whose length is 13 cm.

In an isosceles triangle ABC, if AB = AC = 13 cm and the altitude from A on BC is 5 cm, find BC.

In the given figure, ∠B < 90° and segment AD ⊥ BC, show that

(i) b^{2 }= h^{2 }+ a^{2 }+ x^{2 }- 2ax

(ii) b^{2} = a^{2} + c^{2} - 2ax

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}).

An aeroplane leaves an airport and flies due north at a speed of 1000km/hr. At the same time, another aeroplane leaves the same airport and flies due west at a speed of 1200 km/hr. How far apart will be the two planes after 1 hours?

State the converse of Pythagoras theorem.

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?

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