Home
Class 12
MATHS
Let 'l' is the length of median from the...

Let 'l' is the length of median from the vertex A to the side BC of a `Delta ABC`. Then

A

`4l^(2)=2b^(2)+2c^(2)-a^(2)`

B

`4l^(2)=b^(2)+c^(2)+2bc cos A`

C

`4l^(2)=a^(2)+4bc cos A`

D

`4l^(2)=(2s-a)^(2)"sin"^(2)(A)/(2)`

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D
D is the midpoint of BC

Using Apollonius theorem, we have
`AB^(2)+AC^(2)=2(AD^(2)+BD^(2))`
`rArr c^(2)+b^(2)=2[l^(2)+((a)/(2))^(2)]` (where AD = l)
`rArr 4l^(2)=2b^(2)+2c^(2)-a^(2)`
`=b^(2)+c^(2)+(b^(2)+c^(2)-a^(2))`
`=b^(2)+c^(2)+2bc cos A`
`=(b^(2)+c^(2)-a^(2))+a^(2)+2bc cos A`
`=2bc cos A + a^(2)+2bc cos A`
`=4bc cos A + a^(2)`
Also `4l^(2)=b^(2)+c^(2)+2bc cos A`
`= b^(2)+c^(2)+2bc(1-2"sin"^(2)(A)/(2))`
`=(b+c)^(2)-4bc "sin"^(2)(A)/(2)`
`= (2s-a)^(2)-4bc "sin"^(2)(A)/(2)`
Promotional Banner

Topper's Solved these Questions

  • SOLUTIONS AND PROPERTIES OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Comprehension Type|6 Videos

  • SOLUTIONS AND PROPERTIES OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Comprehension Type|6 Videos

  • SET THEORY AND REAL NUMBER SYSTEM

    CENGAGE PUBLICATION|Exercise Archives|1 Videos

  • STATISTICS

    CENGAGE PUBLICATION|Exercise Archives|10 Videos

Similar Questions

Explore conceptually related problems

Let the lengths of the altitudes drawn from the vertices of Delta ABC to the opposite sides are 2, 2 and 3. If the area of Delta ABC " is " Delta , then find the area of triangle

The lengths of three sides of a triangle are 5cm, 12cm and 13 cm. Then find the length of the perpendicular drawn from the opposite vertex of the side of length 13 cm to that side.

Let f, g and h be the lengths of the perpendiculars from the circumcenter of Delta ABC on the sides a, b, and c, respectively. Prove that (a)/(f) + (b)/(g) + (c)/(h) = (1)/(4) (abc)/(fgh)

If in Delta ABC, b = 3 cm, c = 4 cm and the length of the perpendicular from A to the side BC is 2 cm, then how many such triangle are possible ?

The two sides adjacent to the right-angled triangle are 9 cm and 12 cm. Find the length of the perpendicular drawn from the vertex to the hypotenuse.

A perpendicular AD is drawn on BC from the vertex A of the acute Delta ABC . Prove that AC^(2) = AB^(2) +BC^(2) -2BC.BD OR Prove that the square drawn on the opposite side of theacute angle of an acute triangle is equal to the areas of the sum of the squares drawn on its other two sides being subtracted by twice the area of the rectangle formed by one of its sides and the projection of the other side to this side.

(v) Delta ABC is equilateral and AD is a median of it. If G be the centroid of Delta ABC and GD=10 cm. find the length of GB.

In the triangle ABC,the perpendicular AD, from the point A on the side BC meets the side BC at the point D. If BD=8cm , DC =2 cm andAD = 4 cm, then find the measure of /_ BAC .

AD is a perpendiuclar drawn from the vertex A of the right-angled triangle ABC on its side BC . If AB:AC=5:4 , then find the value of the ratio BD:DC .

ABC is a right angled triangle , right angled at C and P is the length of perpendicular from C on AB. If a, b and C are the length of sides BC, CA and AB respectively. Then___