Area of triangles

and parallelograms

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments.

In Euclidean geometry any three non-collinear points joined together with three line segments is called a triangle…

What is a Triangle?

A

B

C

Nature teaches us….

It is from nature that man has learnt everything and even maths…. Here are some of the triangles that occur in nature….

But man has used his intelligence to apply the shapes of nature ….

In Euclidean geometry, a parallelogram is a convex quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.

What is a parallelogram?

Here are some parallelograms that occur in real life…

Understanding area of figures

Area is a count of how many unit squares fit inside a figure. To fully understand this classic definition of area, we need to picture the unit square. A unit square is a square that is one unit long by one unit wide. It can be 1'x1', 1 m x 1 m, 1 yd x 1 yd, 1" x 1", ...

1

1

Area of triangles

Area of triangle is always calculated by half its base multiplied to its corresponding height…

TriangleArea = ½b × h

b = baseh = vertical height

Area of parallelograms

The area of the parallelogram is got by multiplying its base to its corresponding altitude (height)…

ParallelogramArea = b × h

b = baseh = vertical

height

UNDERSTANDING CONGRUENCE…If two figures have the same shape and the same size, then they are said to be congruent figures…Congruent figures are exact duplicates of each other. One could be fitted over the other so that their corresponding parts coincide.The concept of congruent figures applies to figures of any type.

NATURE TEACHES US…As always , nature is the best teacher.. Nature teaches us congruence by means of reflections and through BILATERAL SYMMETRY etc…Mostly, everything in this world is bilaterally symmetrical especially we, the humans.

CONGRUENCE IN MATHEMATICSIf two figures have the same shape and the same size, then they are said to be congruent figures. For example, rectangle ABCD and rectangle PQRS are congruent rectangles as they have the same shape and the same size.Side AB and side PQ are in the same relative position in each of the figures.We say that the side AB and side PQ are corresponding sides.

DC

BA

R

QP

S

CONGRUENCE IN TRIANGLESCongruent triangles have the same size and the same shape.  The corresponding sides and the corresponding angles of congruent triangles are equal.

A B

C

D E

F

There are five types of congruence rules. They are SSS, SAS, AAS=SAA, ASA and RHS congruence Rules.

SOME THEOREMS RELATED TO CONGRUENCE

If two figures are congruent , they have equal areas.

But its converse IS NOT TRUE.

Two figures having equal areas need not be congruent.

A B

Figures A and B are congruent and hence they have equal areas.

D

CFigures C and D have equal areas, but they are not congruent.

THEOREM 1

Given: Two parallelograms ABCD and EFCD On the same base DC and between the same parallels AF and DC.TO PROVE: ar(ABCD)=ar(EFCD)PROOF: In ADE and BCF,Angle DAE= angle CBF (corresponding angles)Angle AED= Angle BFC (corresponding angles)Angle ADE= Angle BCF (angle sum property)Also, AD =BC (opposite sides of the parallelogram ABCD)Triangle ADE is congruent to triangle BCF (By ASA rule)Therefore, ar(ADE)=ar(BCF) (congruent figures have equal areas)Now, ar (ABCD) = ar(ADE) + ar(EDCB) ar (ABCD) = ar (BCF)=ar(EDCB) ar (ABCD) = ar (EFCD)Hence they are equal in area

D C

A E B F

CONCLUSION OF THE THEOREM 1

Parallelograms on the same base or equal bases and lie between the same parallels are equal in area.

Parallelograms on the same base or equal bases that have equal areas lie between the same parallels.

Parallelograms which lie between same parallels and have equal area lie on the same base.

THEOREM 2

GIVEN : ABCD is a parallelogram, AC is diagonal, Two Triangles ADC and ABC on the same base AC and between the Same parallels BC and AD

TO PROVE : ar (Δ ABC ) = ar (Δ CDA )

Proof: In triangles ABC , CDA

AB = CD (opposite sides of a parallelogram )

BC = DA (opposite sides of a parallelogram )

AC = AC (common )

Therefore Δ ABC Δ CDA (S-S-S congruency)

There fore Δ ABC = Δ CDA (if two triangles are congruent then their areas are equal ).

A B

CD

CONCLUSION OF THEOREM 2

Triangles on the same base and between the same parallels are equal in area.

Triangles on the same base and having equal areas lie between the same parallels.

Triangles between the same parallels and have equal area lie on the same base.

THEOREM 3

IF A TRIANGLE AND A PARALLELOGRAM LIE ON THE SAME BASE AND BETWEEN THE SAME PARALLELS, THEN THE AREA OF THE TRIANGLE IS EQUAL TO HALF THE AREA OF THE PARALLELOGRAM.

Given: triangle ABP and parallelogram ABCD on the same base AB and between the same parallels AB and PC.

To prove: ar(PAB)=1/2 * ar (ABCD) Solution: Draw BQ//AP to get another parallelogram ABQP.

Now parallelograms ABQP and ABCD are on the same base AB and between the same parallels AB and PC

Therefore ar (ABQP) = ar (ABCD) [By theorem 1]……(1) But triangle PAB is congruent to triangle BQP (since

diagnol PB divides parallelogram ABQP into two congruent triangles)

So, ar (PAB) = ar (BQP)…………(2) Therefore ar (PAB) = ½ * ar (ABQP) ………….(3) This gives ar (PAB) = ½ * ar (ABCD)…………..(FROM 1 AND 3)

P D Q C

A B

PRESENTED BY

K.M.Bharthe 9 B

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