Triangles (3) - The right triangle, the Pythagorean Theorem, and other results

*** Basic Points and the Pythagorean Theorem ***

The right-angled triangle (or simply a right triangle) has one right-angle - that is one 90^o  angle

In the next diagram, △ ABC has a right angle at vertex A. Since the three angles of a triangle add to 180^o it follows that the angles at vertices B and C add to 90^o  

The HYPOTENUSE (BC) of the triangle is the longest side and is opposite the right-angle.

If an ALTITUDE is drawn from vertex A to the hypotenuse (at point D), then there are three similar triangles  △ ABC ~ △ BAD ~ △ CAD

Taking the triangles in pairs enables us to write various ratios.

a) Begin with △ ABC ~ △ ABD then, because of similarity,

 =

so AB² = BC.BD

b) Next, take  △ ABC ~ △ ACD

 =

So AC² = BC.CD

Adding

AB² + AC² = BC.BD + BC.CD = BC(BD + CD) = BC²

That is the famous Pythagorean Theorem - the sum of the squares on the legs (AB and AC) of a right-angle triangle is equal to the square on the hypotenuse (BC).

There are numerous proofs of the Pythagorean Theorem - see Cut the Knot - Pythagorean Theorem

 

 *** CIRCUMCIRCLE ***

The circumcircle of a right triangle has the triangle's hypotenuse as its diameter and the midpoint of the hypotenuse is the centre of that circle.

 

*** ACUTE and OBTUSE TRIANGLES ***

ACUTE - An acute triangle has all angles less than  90^o  

OBTUSE - An obtuse triangle has one angle greater than  90^o  

If the side lengths of any triangle are known then it is possible to determine whether the triangle is acute or obtuse.

In the diagram, the triangle to the left is acute. If its side lengths are a, b, c as shown then 

a² + b² > c²

The triangle to the left is obtuse and in this case

a² + b² < c²

 

***A further relationship for OBTUSE triangles ***

In the diagram, it follows from the Pythagorean Theorem that

c² = a² + b² + 2ax

Proof

Apply the Pythagorean theorem to △ ABD

c² = (a + x)² + h² = a² + x² +2ax + h² = a² + (x² + h²) + 2ax

and from △ ADC

b² = x² + h²

and so

c² = a² + b² + 2ax

 

*** APOLLONIUS' THEOREM ***

 

 

Apollonius' Theorem (Wikipedia)