Class 9 Exam  >  Class 9 Notes  >  Extra Documents & Tests for Class 9  >  Procedure - To Show that the Area of a Triangle is half the Product of the Base and the Height, Math

Procedure - To Show that the Area of a Triangle is half the Product of the Base and the Height, Math | Extra Documents & Tests for Class 9 PDF Download

Materials required for real lab:

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

Procedure for Right angled triangle:

As performed in real lab:

  1. Cut a right angle triangle.

  2. Cut a triangle congruent to the right angle triangle.

  3. Align the hypotenuse of the two triangles to obtain a rectangle.[Fig(A)]

Procedure - To Show that the Area of a Triangle is half the Product of the Base and the Height, Math | Extra Documents & Tests for Class 9

As performed in the simulator:

  1. Create a ▲ ABC by providing its three sides. For a right angled triangle with sides a, b, c where c hypotenuse, then a2 + b= c2.

  2. Click on ▲ ABC to create its replica.

  3. 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.

  4. See the observation.

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                                                      

Procedure for Acute Angled Triangle:

As performed in the real lab:

  1. Cut an acute angle triangle and draw the perpendicular from the vertex to the opposite side.

  2. Cut a triangle congruent to it and cut this triangle along perpendicular.

  3. Align the hypotenuse of these cut outs to the given triangle in order to obtain a rectangle.[Fig(B)]

Procedure - To Show that the Area of a Triangle is half the Product of the Base and the Height, Math | Extra Documents & Tests for Class 9

As performed in simulator:

  1. 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.

  2. 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.

  3. Next step is to create two replica triangles of ▲ ABO and ▲ AOC respectively. Click on 'Cut Triangle' button to create these replicas.

  4. 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.

  5. See the observation.

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

For Obtuse Angle Triangle:

As performed in the real lab:

  1. Cut an obtuse angle triangle.

  2. Cut a triangle congruent to this obtuse angle triangle.

  3. Align the greatest side of the two triangles in order to obtain parallelogram.[Fig(C)]

Procedure - To Show that the Area of a Triangle is half the Product of the Base and the Height, Math | Extra Documents & Tests for Class 9

As performed in simulator:

  1. Create a ▲ ABC by providing its three side.

  2. Triangle is Obtuse angle triangle  if its  square of longest side is greater than to sum of products of squares of other two sides.

  3. Click on ▲ ABC to create its replica.

  4. Place this replicated triangle such that it will forms parallelogram.

  5. Use button 'clockwise' to rotate triangle clockwise.

  6. Use button 'Anticlockwise' to rotate triangle Anticlockwise.

  7. Place replicated triangle such that hypotenuse of both triangle will cover each other.

  8. See the observation.

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

The document Procedure - To Show that the Area of a Triangle is half the Product of the Base and the Height, Math | Extra Documents & Tests for Class 9 is a part of the Class 9 Course Extra Documents & Tests for Class 9.
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FAQs on Procedure - To Show that the Area of a Triangle is half the Product of the Base and the Height, Math - Extra Documents & Tests for Class 9

1. How do you calculate the area of a triangle?
Ans. To calculate the area of a triangle, you need to know the length of its base and the height perpendicular to that base. The formula for calculating the area of a triangle is: Area = (1/2) * base * height.
2. What is the significance of the formula for calculating the area of a triangle?
Ans. The formula for calculating the area of a triangle is significant because it provides a simple and straightforward method to find the area of any triangle. By knowing the base and height, you can easily calculate the area without the need for complex measurements or calculations.
3. Can the area of a triangle be negative?
Ans. No, the area of a triangle cannot be negative. The area of any geometric shape represents the amount of space enclosed by that shape, and it is always a positive value. A negative area would indicate a situation where the triangle is flipped or oriented in a way that contradicts its definition.
4. Is the formula for calculating the area of a triangle applicable to all types of triangles?
Ans. Yes, the formula for calculating the area of a triangle is applicable to all types of triangles, regardless of their shape or size. As long as you know the length of the base and the height perpendicular to that base, you can use the formula to find the area of any triangle.
5. Can the formula for calculating the area of a triangle be used for other shapes as well?
Ans. No, the formula for calculating the area of a triangle is specific to triangles only. Other shapes, such as circles, rectangles, or polygons, have their own formulas for finding the area. It's important to use the appropriate formula for each shape when calculating their respective areas.
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