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Principal Properties of Right Triangles


A triangle is a geometric figure which is made of three straight lines and the sum of its angles is equal to 180 degree. A triangle is right triangle or right angled triangle, if any one of its three angles is equal to 90 degree. The side opposite to the 90 degree angle or right angle is known as the hypotenuse.

The properties of right triangle are properties that define a right angled triangle. The properties of right triangles are as follows:

  1. Area of Right Triangle: The right angled triangle area can be measured by using the formula, Area = $\frac{1}{2}$$ \times x \times y$, where ‘x’ and ‘y’ can be considered as the two sides of the triangle. This formula is used only for right angle triangle.
  2. Height or Altitude of Right Triangle: If an altitude from the vertex of the right angle is drawn to the opposite side or the hypotenuse, then two triangles are formed and both the triangles formed are similar to one another and the main triangle.
  3. The pythagoras theorem states that, if ‘h’ be the hypotenuse and ‘x’ and ‘y’ be the two sides of the triangle, then according to the pythagoras theorem, it is stated that $h^2 = x^2 + y^2$. So, according to the formula, hypotenuse’s square is equal to the sum of the square of other two sides of the triangle.
  4. Radius of Incircle and Circumcircle of Right Angled Triangle: According to this property, the radius of the incircle of the triangle can be found using the formula r = $\frac{x + y - h}{2}$, where ‘x’ and ‘y’ are the two sides of a triangle and 'h' is the hypotenuse. The radius of the circumcircle of a right angle triangle can be calculated using the formula r = $\frac{h}{2}$, where ‘h’ is the hypotenuse of the right angled triangle.

Area of Right Triangle

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We know that polygons are closed figures made up of just line segments. Now, all closed figures enclose some region within it. This enclosed region inclusive of the boundary of the polygon is called the area of the polygon. Area of any polygon is measured in terms of the squares of 1 unit each, that can accommodate in the figure. Now, polygons being rectilinear figures, it becomes rather easier to divide them into such unit squares.

As polygons are made up of just line segments, depending on the number of sides each polygon has, they are put under different categories. The polygons with 3 sides are called triangles, those with 4 sides are quadrilaterals,those with 5 sides are pentagons and so on.

A broad classification of each type is whether a polygon is regular with all sides equal or otherwise irregular. But of all the types of polygons, the triangles and the quadrilaterals are further classified into different groups. Let us learn about triangles first.

The triangles are classified on the basis of measure of their angles, as well as, on the basis of the measure of their sides. We shall classify them first on the basis of their angles. On this basis, the triangles may be classified as follows:
  1. Acute triangle
  2. Right triangle
  3. Obtuse triangle
Coming back to area, all polygons enclose area. So, triangles also have some area. Let us try to understand area of a right triangle. We define area of right triangle as the region enclosed by a right triangle inclusive of its boundary. Area right triangle = $\frac{1}{2}$$ \times \text{base} \times \text{height}$, where base and height are the two sides of the triangle containing the right angle.

Inradius and Circumradius of Right Triangle

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Circular geometry contains different kind of radius like inradius and circumradius. Inradius is a radius, which is the part of incircle or insphere and circumradius is a radius, which is part of circumcircle or cicumsphere.

We use the following steps for evaluating an inradius and circumradius:
Step 1: First, we have to check what kind of polygon we have, like regular polygon or irregular polygon.
Step 2: If we have regular polygon, then we use following formula for evaluation of inradius and circumradius:

Inradius of regular polygon = $\frac{1}{2}$$ \times a \times \cot $$\frac{\pi}{n}$,

Here, a is side length and n is number of side of polygon.
Circumradius of regular formula = $\frac{1}{2}$$ \times a \times \csc $$\frac{\pi}{n}$

Here, ‘a’ is the side length and n is the number of sides of polygon.

Step 3: If we have irregular polygon, then we should calculate inradius and circumradius of polygon manually by their graphic structure.

Altitude of Right Triangle

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Dropping of a geometry altitude is a special case of orthogonal projection. An altitude in geometry is the shortest distance from the top or vertex of a geometric figure to the opposite side or base of that geometric figure.

Geometry altitude of a triangle is nothing but a straight line through the top point or vertex and it makes a right angle or is perpendicular with a line opposite of the triangle or base. The line which contains the opposite side of any geometric figure is called the extended base of the geometry altitude. The point where the intersection takes place between the extended base and the altitude in geometry is called the foot of the altitude. The distance between the base and the vertex of any geometric figure is called the length of the altitude or simply altitude in geometry. We can draw an altitude in geometry from the vertex to the foot of a geometric figure by a process known as dropping.

Three geometry altitude of a triangle intersect at a single point called an orthocenter.

Geometry altitude can be used to calculate the area of a triangle by using the formula $\frac{1}{2}$ bh, where b is the length of the base and h is the length of the altitude or height of the triangle. The altitudes of a triangle are related to the sides of the triangle through many theorems like equilateral triangle theorem, inradius theorems, area theorem and so on. There is one very important formula called Heron’s formula which relates to the sides and altitude of a triangle.

In an isosceles triangle, the foot of the geometry altitude is the midpoint of the base of the triangle. And, the altitude will form the angle bisector of the vertex of the triangle.