Maths help......Triangles.?

Question

In a triangle, ABC angle A is 90 degree and D is the mid point of AC. the value of BC^2 - BD^2 = ?

(a)AD^2

(b)2AD^2

(c)4AD^2

(d)3AD^2

Explain.Thank You.

Answer 1

draw the triangle.

NOW D IS THE MIDPOINT OF AC.

JOIN BD

SO AD = DC

THUS AC = AD+DC  ==> AC = 2AD  ........(i)

USING PYTHAGORAS' THEOREM,

BC^2 = AB^2 + AC^2

ALSO, BD^2 = AB^2 + AD^2

BC^2 - BD^2 = AC^2 - AD^2 = 2AD^2 - AD^2  ..........FROM (i)

= AD^2.                  

THEREFORE (A) IS THE ANS.

Answer 2

Using pythogoras theorem in triangles BAD and BAC, we get

in triangle BAD

BD^2 = AB^2 + AD^2 ______(1)

In triangle BAC,

BC^2 = AB^2 + AC^2 _______(2)

subtracting (1) from (2)

BC^2 - BD^2 = AC^2 - AD^2

Now AC = 2AD (since D is the midpoint)

So,

BC^2 - BD^2 = 4AD^2 - AD^2

= 3AD^2

therefore the correct answer is D.

Answer 3

   since it is a rt angled triangle

   so  BC^2 = AB^2 + AC^2--------- 1

   and given D is mid point of AC therefore ADB is a rt angled triangle

   hence BD^2 = AB^2 + AD^2 --------- 2

   now given

    BC^2 - BD^2

    AB^2 + AC^2  -  BD^2  (from  Eq.1)

    AB^2 + AC^2  -  AB^2 - AD^2  (from  Eq.2)

    AC^2 - AD^2   (  cancel AB^2 )

    4 AD^2 - AD^2    since d is mid point of AC  AC^2 = (2 AD)^2

    3 AD^2

      

Answer 4

AB =x , AD = y, AC = 2y

BC^2 = x^2 + (2y)^2

BD^2 = x^2 + y^2

Simple

Ans : 3AD^2

Answer 5

b)2AD^2

Answer 6

3 AD *2 is the right answer

Answer 7

The correct answer is d.

Use pythogoras theorem on two triangles BAD and BAC.

in triangle BAD

BD^2 = AB^2 + AD^2 (1)

In triangle BAC,

BC^2 = AB^2 + AC^2 (2)

Now subtract (1) from (2)

BC^2 - BD^2 = AC^2 - AD^2

Now AC = 2AD (since D is the midpoint)

So,

BC^2 - BD^2 = 4AD^2 - AD^2

                   = 3AD^2

Answer 8

Hi,

Let AB = 3, AC = 4, and BC = 5 ( since 3² + 4² = 5²)

If AC = 4 and D is the midpoint of AC, then AD = 2 and DC = 2

To find BD², BD² = BA² + AD²

BD² = BA² + AD²

BD² = 3² + 2²

BD² = 9 + 4

BD² = 13

BC² - BD² =

5² - 13 =

25 - 13 = 12

Since AD = 2, then

(a)AD² = 2² = 4

(b)2AD² = 2*2² = 8

(c)4AD² = 4*2² = 16

(d)3AD² = 3*2² = 12 <==ANSWER

BC² - BD² = 3AD² <==ANSWER

I hope that helps!! :-)

Answer 9

Its evident ABD and ABC are two right angled triangles

IN Triangle ABD,

BD^2 = AB^2 + AD^2

In Triangle ABC

BC^2 = AB^2 + AC^2

BC^2 - BD^2 = AC^2 - AD^2

Now D is mid point of AC, so AD = DC = 1/2 AC

So BC^2 - BD^2 = 4AD^2 - AD^2 = 3 AD^2

Answer 10

In triangle ABC, BC² = AC² + AB² --> (1) (Pythagoras Theorem)

Similarly

In triangle ABD, BD² = AB² + AD² --> (2)  (Pythagoras Theorem)

(1)-(2) Gives: BC² - BD² = AC² - AD² --> (3)  

Since D is the mid-point of AC, AC = AD +DC= 2AD (Since D is the midpoint of AC, AD = DC)

Hence AC² = 4AD²

Applying this in (3), we get BC² - BD² = 4AD² - AD² = 3AD²

Hence (d) is the answer.

AJM