Lesson Right-angled triangle

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In this lesson we will look at the basic definition and properties of the right angled triangle.

Right Angled Triangle
'Right Angled Triangle' is a <A HREF=http://www.algebra.com/tutors/Triangles_and_its_basic_properties.lesson?content_action=show_dev>triangle</A> with one internal angle equal to 90 degrees (<A HREF=Right_angle.wikipedia>right angle</A>). 
The side opposite to the right angle is called "<A HREF=Hypotenuse.wikipedia>hypotenuse</A>" and hypotenuse is the longest 
side of the right angled triangle. The other two sides adjacent to the right angle are called 
legs or catheti.

{{{drawing( 160, 160, -0.5, 7, -1, 4.5, line( 0, 0, 6, 0 ), line( 0, 0, 0, 4 ),line( 6, 0, 0, 4), locate( 0, 0, A ),locate( -.5, 4.3, C ),locate( 6, 0, B ),locate( 2.5, 0, c(leg) ),locate( 0.2, 2, b(leg) ),locate( 3, 2.7, a(hypotenuse) ))}}}

Properties of Right Angled Triangle

1. If any two side lengths are given then we can find the third side length by famous 
"<A HREF=Pythagorean_theorem.wikipedia>Pythagorean theorem</A>". i.e. If we let 'a' be the length of the hypotenuse and 'b' and 'c' 
be the lengths of the other two sides, the theorem can be expressed as the equation.

{{{a^2=b^2+c^2}}}

2. Acute angles of the right angled triangle are complimentary. i.e. Sum of the two acute angles is 90 degrees.

3. If both acute angles are same then the both legs are of equal length and vice-versa. Hence both acute angles are 45 degrees.

Types of Right Angled Triangle

We can categorized the right angled triangle into three categories.

(a) 30-60-90 triangle

In 30-60-90 triangle, angles are 30 degree, 60 degrees and 90 degrees.

{{{drawing( 160, 160, -0.5, 7, -0.5, 5.5, line( 1, 0, 4, 0 ), line( 1, 0, 1, 5.196 ),line(1, 5.196, 4, 0), locate( 1, 0, A ),locate( .5, 5.4, C ),locate( 4, 0, B ),locate( 3, 2.7, a ),locate( 2.5, 0, c ),locate( .5, 2.7, b ),locate( 3, .6, 60 ),locate( 1.1, 4, 30 ))}}}

Relation among the side lengths:

If 'a' is the length of the hypotenuse then we can find the length of two legs 'b' and 'c' 
in terms of the length of the hypotenuse as follows:

1) {{{b = sqrt(3)*(a/2))}}}

By using <A HREF=Sine_rule.wikipedia>Sine Rule</A> at angle B:

{{{b = a* Sin(60)}}}
{{{b = sqrt(3)*(a/2))}}}

2) {{{c = a/2}}}

By using <A HREF=Sine_rule.wikipedia>Sine Rule</A> at angle C:

{{{c = a*Sin(30)}}}
{{{c = a/2}}}

(b) Isosceles right angled triangle

In Isosceles right angled triangle, one right angle and acute angles are of 45 degrees.

{{{drawing( 160, 160, -0.5, 7, -0.5, 5.5, line( .5, 0, 5, 0 ), line( .5, 0, .5, 4.5 ),line(.5, 4.5, 5, 0), locate( .5, 0, A ),locate( .5, 5.4, C ),locate( 5, 0, B ),locate( 3, 2.7, a ),locate( 2.5, 0, c ),locate( 0, 2.7, b ),locate( 3.1, .6, 45 ),locate( .6, 3.8, 45 ))}}}

Relation among the side lengths:

If 'a' is the length of the hypotenuse then we can find the length of two legs 'b' and 'c' 
in terms of the length of the hypotenuse as follows:

Both legs are of equal length

{{{b=c= a*Sin(45)}}}
{{{b=c=a*(1/sqrt(2))}}}

(c) Scalene right angled triangle

In Scalene right angled triangle, one right angle and other two angles are not equal.
No two sides are equal in Scalene right angled triangle.
30-60-90 triangle is a particular case of Scalene right angled triangle.

{{{drawing( 160, 160, -0.5, 6.5, -0.5, 6.5, line( 1, 0, 6, 0 ), line( 1, 0, 1, 3 ),line(1,3, 6, 0), locate( 1, 0, A ),locate( 1, 3.6, C ),locate( 6, 0, B ),locate( 4, 2.3, a ),locate( 2.5, 0, c ),locate( .5, 1.8, b ))}}}

For further reading on Right Angled Triangle 
<A HREF=Right_angled_triangle.wikipedia>Wikipedia</A>.