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6.3 Special Right Triangles

The 300 - 600 - 900 triangle : If the angles of a triangle are 300 , 600 and 900 then the side opposite to 300 is half the hypotenuse and the side opposite to 600 is times the hypotenuse.

Figure 6.5

In D ABC Ð A = 900 Ð B = 600 and Ð C = 300

To prove that l (AB) = l (BC) and l (AC) =

Take a point D on ray BA such that seg.AD @ seg.AB and join CD.

In D ABC and D ACD

AB @ AD construction

Ð CAB @ Ð CAD both are right angles

AC @ AC common side

\ D BAC @ D DAC (SAS)

\ Ð ACB @ Ð ACD corresponding angles of congruent triangles are equal.

but m Ð ACB = 300

\ m Ð DCB = 600

Þ D DCB is an equilateral triangle

\ l (seg.DC) = l (seg.BC) = l (seg.DB) ® (1)

but l (seg. AB) = l (DB) ® (2)

®(3) from (1) and (2).

In right triangle ABC

l (seg.AB)2 + l (seg.AC)2 = l (seg.BC)2 Pythagoras

{ l (seg.BC)2 } + l (seg.AC)2 = l (seg.BC)2from (3)

l (seg.AC)2 = l (seg.BC)2 - l (seg.BC)2

= l (BC)2

The 450 - 450 - 900 triangle : If the angles of a triangle are 450 - 450 - 900 then the perpendicular sides are times the hypotenuse.

In D ABC Ð A = 450 , Ð B = 900 and Ð C = 450.

Figure 6.6

To prove that AB = BC =

By Pythagoras theorem

l (seg.AB)2 + l (seg.BC)2 = l (seg.AC)2

l (seg.AB) = l (seg.BC) . D ABC is isosceles.

\ l (seg.AB)2 + l (seg.BC)2 = 2 l (seg.AB)2 = l (seg.AC)2

Index

6.1 The Right Triangle
6.2 The Theorem of Pythagoras
6.3 Special Right Triangles

Chapter 7

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