Right Triangle

A right triangle is a polygon with three sides that has one angle (α) that measures 90° which is the largest angle of the right triangle.

If we add all three angles in any triangle we get 180 degrees. Thus, the sum of the other two smaller angles is 90°.

The relation between the sides lengths and angles of a right triangle is the basis for trigonometry.

The elements of a right triangle are:

• Legs: the two sides contiguous to the right angle, a and b (each of them is a leg), and the
• Hypotenuse: the long side c, opposite the right angle (the largest angle).

Types of Right Triangles

There are two types of right triangles depending on their two acute angles:

• Isosceles Right Triangle: triangle with one right angle (90°) and two other equal angles (45°). The two legs are equal in length.



• Scalene Right Triangle: all three angles have different measures (one of them measures 90°). Also its sides have different lengths.

Special Right Triangle

A special right triangle is a right triangle whose sides or angles are in a particular ratio. There are some special right triangles so common that it’s useful to know its sides ratios. This allows you to find the missing sides when you only know one side without resorting to more advanced methods like the Pythagorean Theorem.

Special right triangles fall into two categories:

1. Angle-Based Special Right Triangle.There are two angle-based special right triangle:
   • 

     45-45-90 Right Triangle: as we saw above this is the isosceles right triangle whose sides ratios are x, x, x√2. In other words, the sides are in the ratio 1:1:√2 and angles are in the ratio 1:1:2.

   • 

     30-60-90 Right Triangle: is a special scalene right triangle whose angles measure 30-60-90. They are in a ratio 1:2:3. A 30-60-90 right triangle has side ratios x, 2x and x√3 (or ratio 1:2:√3)
2. Side-Based Special Right Triangle. There are special right triangles whose sides are of integer lengths. This type of right triangle is called Pythagorean Triple Triangle.

   The most common Pythagorean Triple Triangles are whose sides are in the ratios:

   • 3: 4 :5
   • 5: 12 :13
   • 8: 15 :17
   • 7: 24 :25
   • 9: 40 :41

   We can find more Pythagorean Triples Triangles by scaling any other Pythagorean number Triple. For example multiplying with 3 the Pythagorean triple 3:4:5 we obtain 9:12:15.

Altitude of a Right Triangle

In a right triangle the altitude of each leg (a and b) is the corresponding opposite leg. Thus, h_a = b and h_b = a. The altitude of the hypotenuse is h_c.

The three altitudes of a triangle intersect at the orthocenter H which for a right triangle is in the vertex C of the right angle.

To find the height associated with side c (the hypotenuse) we use the geometric mean altitude theorem.

We can calculate the altitude h (or h_c) if we know the three sides of the right triangle.

How to find the Area of a Right Triangle

A right triangle has one right angle (90°), so its height coincides with one of its sides (a). Its area is half the product of the two sides that form the right angle (legs a and b).

How to find the Perimeter of a Right Triangle

To find the perimeter of a right triangle we add the lengths of its three sides.

Another way to calculate the perimeter of a right triangle is applying the Pythagoras’ theorem. This theorem states that the square of the two legs on a right triangle is equal to the square of the hypotenuse. So, we can find the perimeter of a right triangle if we have the length of the two legs (a and b) cause hypotenuse is equal to √(a²+b²).

Remember that Pythagorean Theorem states:

Then the perimeter is:

The Pythagorean Theorem

The Pythagorean Theorem, also known as Pythagoras’s theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle (2 legs and hypotenuse). This theorem can be written as the following equation:

Geometric Mean Theorem

The Geometric Mean Theorem (or Altitude-on-Hypotenuse Theorem) relates the height (h) of the triangle and the legs of two triangles similar to the main ABC, by plotting the height h over the hypotenuse, stating that in every right triangle, the height (h) relative to the hypotenuse is the geometric mean of the two projections of the legs on the hypotenuse (n and m).

Leg Rule

The leg Rule is a theorem that relates the segments projected by the legs on the hypotenuse with the legs they touch.

In every right triangle, a leg (a or b) is the geometric mean between the hypotenuse (c) and the projection of that leg on it (n or m).

Thales’ Theorem

Thales’ Theorem is a special case of the inscribed angle theorem, it’s related to right triangles inscribed in a circumference.

Thales’ theorem states that if A, B, and C are distinct points on a circle with a center O where the line AC is a diameter, the triangle Δ ABC has a right angle (90°) in point B. Thus, Δ ABC is a right triangle.

In other words, the diameter of a circle always subtends a right angle to any point on the circle.

Proof of Thales’ Theorem

If we connect the center O to point B we create two triangles Δ ABO and Δ OBC that are both isosceles triangles because all radii r are equal (OA, OB and OC are equal). And, by the base angle theorem, their base angles are equal. Let’s label the base angles of Δ ABO ‘α’, and those of Δ OBC ‘β’.

As in any triangle, the interior angles of the triangle Δ ABC add up to 180°:

Dividing the equality by 2:

Since α + β is the angle of Δ ABC at point B, Thales’ theorem is proved.

Solved Exercises

Exercise

Find the area of a right triangle given its two legs, which form the right angle: a = 3 cm and b = 4 cm.

Solution:

Apply the above formula:

And we have that the area is 6 cm².

Exercise

We can find the perimeter of a right triangle whose sides are a=3 cm, b=4 cm and c=5 cm by adding all the three sides:

The perimeter of the triangle is 12 cm.