Area of triangle

Chart paper, pencil, compass, scale, a pair of scissors, cello tape.

- Cut a right angle triangle.
- Cut a triangle congruent to the right angle triangle.
- Align the hypotenuse of the two triangles to obtain a rectangle.[Fig(A)]

- Create a ▲ ABC by providing its three sides. For a right angled triangle with sides a, b, c where c hypotenuse, then a
^{2}+ b^{2 }= c^{2}. - Click on ▲ ABC to create its replica.
- Place replicated triangle such that hypotenuse of both triangle will cover each other. Use button 'clockwise' to rotate triangle clockwise. Use button 'Anticlockwise' to rotate triangle Anticlockwise.
- See the observation.

We can observe that two congruent triangles aligned on hypotenuse forms a rectangle.

∴ Area of □ ABCD = 2 x Area of ▲ ABC

∴ Area of ▲ ABC = 1/2 x Area of □ ABCD

= 1/2 x [base of □ ABCD X height of □ ABCD]

= 1/2 x [BC x AB]

= 1/2 x base of triangle ABC x height of triangle ABC=1/2 x base x height

- Cut an acute angle triangle and draw the perpendicular from the vertex to the opposite side.
- Cut a triangle congruent to it and cut this triangle along perpendicular.
- Align the hypotenuse of these cut outs to the given triangle in order to obtain a rectangle.[Fig(B)]

- Create a ▲ ABC by providing its three sides. Triangle is Acute angled triangle if its square of longest side is less than to sum of products of squares of other two sides.
- Next step is to draw perpendicular from A To line BC.
- Click on SetSquare in the 'Tools' to use it.
- Drag this set square and place at position such that point A will perpendicular to base BC.

- Next step is to create two replica triangles of ▲ ABO and ▲ AOC respectively. Click on 'Cut Triangle' button to create these replicas.
- Next step is to place these colored triangles at appropriate positions.
- First you have to place yellow colored triangle and then red colored triangle.
- Drag yellow colored triangle and place along with its hypotenuse to side AB of ▲ AOB which finally forms a rectangle AOBD.
- Drag Red colored triangle and place along with its hypotenuse to side AC of ▲ AOC which finally forms a rectangle AOCE.
- You can use 'clockwise' button to rotate triangle clockwise.
- You can use 'Anti-clockwise' button to rotate triangle Anti-clockwise.

- See the observation.

As □ DBCE is formed with ▲ ABC and 2 congruent triangles ABO and AOC.

∴ Area of □ DBCE = Area of ▲ ABC + (Area of ▲ ABO + Area of ▲ AOC)

= Area of ▲ ABC + Area of ▲ ABC

= 2 x Area of ▲ ABC

Area of ▲ ABC = 1/2 x Area of □ DBCE

= 1/2 x [base of □ DBCE X height of □ DBCE]

= 1/2 x [BC x DB] = 1/2 x [BC x AO]

= 1/2 x base of triangle ABC x height of triangle ABC = 1/2 x base x height

- Cut an obtuse angle triangle.
- Cut a triangle congruent to this obtuse angle triangle.
- Align the greatest side of the two triangles in order to obtain parallelogram.[Fig(C)]

- Create a ▲ ABC by providing its three side.
- Triangle is Obtuse angle triangle if its square of longest side is greater than to sum of products of squares of other two sides.
- Click on ▲ ABC to create its replica.
- Place this replicated triangle such that it will forms parallelogram.
- Use button 'clockwise' to rotate triangle clockwise.
- Use button 'Anticlockwise' to rotate triangle Anticlockwise.
- Place replicated triangle such that hypotenuse of both triangle will cover each other.
- See the observation.

You can observe that aligning these two congruent triangles forms a parallelogram.

As per property of parallelogram ▲ ABC and ▲ ADC are congruent

∴ Area of ▱ ABCD = Area of ▲ ABC + Area of ▲ ADC

= 2 x area of ▲ ABC

∴Area of ▲ ABC = 1/2 x Area of ▱ ABCD

= 1/2 x [base of ▱ ABCD X height of ▱ ABCD]

= 1/2 x [BC x height of ▲ ABC]

= 1/2 x base of ▲ ABC x height of ▲ ABC = 1/2 x base x height