You know this… A triangle with one right angle is called a right triangle. The side opposite the right angle is called the hypotenuse of the triangle. The other two sides are called legs. The other two angles are always acute and complementary.
The triangle above has side c as its hypotenuse, sides a and b as its legs. Angle C is its right angle. Angles A and B are acute and complementary.
There are two types of right triangles that every geometry student should know very well.
One is the right triangle formed when an altitude is drawn from a vertex of an equilateral triangle, forming two congruent right triangles.
The angles of the triangle will be 30, 60, and 90 degrees, giving the triangle its name: 30-60-90 triangle.
The ratio of side lengths in such triangles is always the same: if the leg opposite the 30 degree angle is of length x, the leg opposite the 60 degree angle will be of x
, and the hypotenuse across from the right angle will be 2x.
The other common right triangle results from the pair of triangles created when a diagonal divides a square into two triangles.
Each of these triangles is congruent, and has angles of measures 45, 45, and 90 degrees.
If the legs opposite the 45 degree angles are of length x, the hypotenuse has a length of x
. This ratio holds true for all 45-45-90 triangles. 45-45-90 triangles are also often called isosceles right triangle.
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45°- 45°- 90° Triangle – a right triangle where the angles are 45°, 45°, and 90°.
45-45-90 Triangle Theorem – In a 45-45-90 triangle, the hypotenuse is times as long as each leg.
The side-length ratios of leg: leg: hypotenuse are 1 : 1 : .
This is one of the ’standard’ triangles you should be able recognize on sight.
A fact you should commit to memory is: The sides are always in the ratio 1 : 1 : . With the ‘‘ being the hypotenuse (longest side). This can be derived from Pythagoras’ Theorem. This ratio will come in handy later in the study of trigonometry.
Because the base angles are the same (both 45°) the two legs are equal and so the triangle is also isosceles.
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30°- 60°- 90° Triangle – a right triangle where the angles are 30°, 60°, and 90°.
30-60-90 Triangle Theorem – In a 30-60-90 triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is 
times as long as the shorter leg.
The side-length ratios of short leg: long leg: hypotenuse are 
This is one of the ’standard’ triangles you should be able recognize on sight. A fact you should commit to memory is: The sides are always in the ratio 1 : 
: 2. With the ‘2′ being the hypotenuse (longest side). This ratio will come in handy later in the study of trigonometry. Also notice that the smallest angle is always opposite the smallest side.
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The Wheel of Theodorus.
Theodorus of Cyrene was the tutor of Plato, teaching him mathematics. He was a member of the society of Pythagoras.
