Triangles : Construction and Properties - Part I

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A triangle is a three-sided figure. Study of triangles is one of the most basic studies in geometry. It introduces you to the use of the compass or geometry box and opens up a lot of concepts in geometry. 

What we will study in this chapter :

1. Construction of a triangle
2. Properties of a triangle
3. Types of triangles
4. Other definitions 

1. Construction of a triangle
A triangle has three sides and three angles i.e. there are a total of 6 parameters. For correct construction of a triangle, at least 3 of the 6 quantities have to be given.  The three points of a triangle are also known as the vertices of the triangle.

Points A, B and C are the vertices of the triangle ABC (written as ABC).

Segments AB, BC, AC are called the sides of  the ABC.

Angle at point A is written as A or CAB; similarly other angles B or C are also written in the same way.

It has been known that there are only three methods that lead to a unique way for drawing a triangle.  They are :

  • Three sides are given or SSS construction

  • Two sides with the angle included by the two given sides or SAS construction

  • Two angles with the side joining the two angles or ASA construction.

(i) SSS construction : In this construction the measurements of the three sides or segments are given.

Example : Construct ABC  such that l(AB) = 5cm, l(BC) = 4 cm and l(AC) = 3cm.

First take a ruler and draw a segment AB = 5 cm. Write A at the beginning of the segment and B at the end of the segment.

Then take the compass from your geometry box and measure a length of 3cm with the help of a ruler. Place the compass point on A and draw an arc of a circle with the compass.

Next measure a length of 4cm with the compass and draw another arc from point B.

The arcs drawn from point A and point B should intersect. The intersection point is point C of the triangle.

Now complete the triangle by joining point A to C and point B to C.

ABC is completely constructed.

(ii) SAS construction : In this construction the measurements of the two sides or segments are given, the angle enclosed within the two sides is also given.

Example : Construct PQR  such that l(PQ) = 4cm, l(QR) = 4.5 cm and Q or PQR = 60ƒ.

First take a ruler and draw a segment PQ = 4 cm. Write P at the beginning of the segment and Q at the end of the segment.

Take the protractor from your geometry box and place the center of the protractor on point Q. Mark an angle of 60ƒ by drawing a sharp point. Draw a ray starting from Q and joining the point, so that the ray makes an angle of 60ƒ with respect to PQ.

Take the compass and measure a distance of 4.5 cm. Place the compass at point Q. Then draw and arc of a circle. Where the arc intersects the ray from Q is the point R of the triangle.

Now join points P and R, Q and R.  

PQR is completely constructed.

(iii) ASA construction : In this construction the measurements of the two angles are given, the side enclosed within the two angles is also given.  

Example : Construct XYZ  such that X or YXZ = 60ƒ, Y or XYZ = 35ƒ, and l(XY)= 3cm.

First take a ruler and draw a segment XY = 3 cm. Write X at the beginning of the segment and Y at the end of the segment.

Take the protractor from your geometry box and place the center of the protractor on point X. Mark an angle of 60ƒ by drawing a sharp point. Draw a ray starting from X and  joining the point, so that the ray makes an angle of 60ƒ with respect to XY.

Then place the center of the protractor on point Y. Mark an angle of 35ƒ by drawing a sharp point. Draw a ray starting from Y and joining the point, so that the ray makes an angle of 35ƒ with respect to XY.

Where the two rays, one starting from point X and the other starting from point Y meet, is the point Z of the triangle.

Now join points X and Z, Y and Z.

XYZ is completely constructed.

2. Properties of a triangle  
There are basically two properties which are connected with the study of triangles : 

1. The sum of all the three angles = 180ƒ.

2. Area of the triangle = 1/2 * base * height

Base of the triangle is defined as any of its sides. 

From the vertex opposite to the base, drop a perpendicular to the base. The length of this perpendicular is known as the height or altitude of the triangle.

In the ABC shown, if we consider side AB to be the base, then draw a perpendicular from vertex C to side AB.  This is shown by the line CD. CD is the height of the triangle. Segment CD is  to segment AB. Measure the length AB and CD with a ruler. Area of the ABC is half of the base (AB) multiplied by the height (CD).

Instead of base as AB, we can take the base to be either AC or BC. Then the height has to be determined by drawing a perpendicular from vertices B or A respectively. The result will be the same.  (For obtuse angled triangles (defined later), the perpendiculars will fall outside the triangle).

Area of a triangle can also be calculated from its perimeter.  A perimeter is the sum of all sides. If we denote the perimeter to be S and  l(AB) = a, l(BC) = b, l(AC) = c,

Then S = a + b + c, and the semi-perimeter  s =  S/2  =  1/2  (a + b + c)  

                                         
The area of the triangle  =   

 

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