IX Proof of Heron’s formula

IX Proof of Heron’s formula

Let a, b, c are length of the sides and h is height to side of length c  of ∆  ABC.

We have S = (a + b + c)/2

So, 2s = a + b + c 

 Þ  2(s - a) = - a + b + c 
Þ  2(s - b) = a - b + c 
Þ  2(s - c) = a + b – c

Let p + q = c as indicated.

Then,   h² =  a² - p²       -------------(1)

 Also,   h² = b² - q ²       -------------- (ii)

From (i) and (ii)

  Þ  a² - p^2     =   b² -  q ²       

   Þ  q²   = - a² + p² + b²      

Since,   q = c - p  Þ q² = (c-p)² Þ q²  = c² + p² -2pc

Then, c² + p² -2pc   = - a² + p² + b²      

Þ - 2pc  =  - a² +b² – c² = - ( a² -b² + c²)

Þ   p = ( a² -b² + c²)/2c

Now, Put this value of p in equation (i)

h²    =   a² -  p²     

h²      =   ( a – p ) ( a + p )

h²   =    {a – ( a² -b² + c²)/2c }   {a + ( a² -b² + c²)/2c }

h²   =   {(2ac - a² + b² - c²)/2c}x{(2ac+ a² - b² + c²)/2c}

h²      = {(b² – (a - c)² }{(a + c)² – b²}/4c²

h²     = {(b – a + c) (b + a - c)}{(a + c + b)(a + c – b )

h² = { 2(s - a) x 2(s - c)  x 2s x2(s - b)}/4c²

h² = { 4 s (s - a) x (s - c) x(s - b)}/c²

h = 2/c  √ s (s - a) x (s - b) x(s - c) 

½ h c = √ s (s - a) x (s - b) x(s - c) 

Area of triangle = √ s (s - a) x (s - b) x(s - c)

CBSE Test paper-1

1. Two sides of a triangle are 8cm and 11cm and its perimeter is 32cm.The third side is :

(a) 4cm (b) 13cm (c) 14cm (d) 16cm

2. The base of a triangle is 12cm and height is 8cm .Its area is:

(a) 24cm2 (b) 96cm2 (c) 48cm2 (d) none

3. The sides of a triangular plot are in the ratio 3:5:7 and its perimeter is 300m . The sides of a triangle are.

(a) 60m,100m,40m (b) 50m,80m,60m (c) 45m,75m,95m (d) none

4. What will be the area of quadrilateral ABCD if AB =3cm, BC=4cm, CD=4cm, DA=5cm and AC=5cm.

(a) 12.5cm (b) 15.2cm (c) 18.2cm (d)19.2cm

5. An isosceles triangle has perimeter 30cm and each of equal side is 12cm .Area of triangle is:

(a) 8√15cm2 (b) 7√12cm2 (c) 9√15cm2 (d)none

Complete the following sentences

6. Area of an equilateral triangle with side ‘a’ is _______________.

7. If a, b, and c are the three sides of a triangle then by Hero’s formula area is___________.

8. In Heron’s formula semi perimeter is equal to ____________.

9. Area of a right angled triangle is ________________.

10. The area of a parallelogram is 392m2.If its altitude is twice the corresponding base, determine the base and height.

11. The adjacent sides of a parallelogram are 36cm and 27cm in length .If the distance between the shorter sides is 12cm, find the distance between the longer sides.

12. A rectangular lawn, 75m by 60m, has two roads , each 4m wide, running through the middle of the lawn, one parallel to length and other parallel to breadth. Find the cost of gravelling the roads at Rs 5.50 per m2

13. Using Heron’s formula, find the area of an equilateral triangle if its side is ‘a ‘units.

14. Find the percentage increase in the area of a triangle if its each side is doubled.

15. Find the area of quadrilateral ABCD whose sides in meters are 9, 40, 28 and 15 respectively and the angle between first two sides is a right angle.

16. The difference between the sides containing a right angle in a right angled triangle is 14cm. The area of a triangle is 120cm2.Calculate the perimeter of a triangle.