A triangle is a three-sided, three angled polygon where the sum of internal angles is always 180 degrees. In a triangle, altitude is the line that begins from the vertex, extends to the opposite side of the triangle and forms a right angle with that side of the triangle. The geometrical properties of altitudes in different types of triangles are provided here. The high school students can learn the interesting topic about triangles from the following sections.
Geometrical Property of Altitudes
The altitude of a triangle is defined as the perpendicular line segment that is drawn from the vertex of the triangle to its opposite side. So, the altitude makes a right angle to the side of the triangle. Commonly, it is called the height of the triangle.
At times the opposite side is not enough long to draw an altitude, so we have to extend it to make an altitude possible. This line having the opposite side is called the extended base of the altitude. The point of intersection of altitude and the opposite side is called the foot of the altitude. The geometrical property of altitudes is along the lines:
- If three altitudes of a triangle are concurrent. The point at which they meet is called the orthocentre of the triangle.
- In a scalene triangle, none of the altitudes has the same length.
- In an isosceles triangle, any two altitudes have the same length.
- In an equilateral triangle, all three altitudes have the same length.
- All the altitudes are inside the triangle for an acute triangle.
- The altitude drawn from the obtuse vertex is inside the triangle, the other two altitudes connected to the acute vertices lie outside the obtuse triangle.
- In a right triangle, the altitude perpendicular to the hypotenuse is inside the triangle and others are legs of the triangle.
Altitudes of Triangle – Definition, Properties
The altitude of a triangle is the line segment that joins vertex, opposite side at right angles. If all the altitudes meet at one point is called orthocenter O.
Properties of Altitude of Triangle:
- All triangles have exactly 3 altitudes each from one vertex.
- Altitude is the shortest distance from the vertex to its opposite side.
- The triangle altitude may lie inside or outside the triangle.
- The 3 altitudes always meet at a single point irrespective of triangle shape. That point is called the orthocenter of the triangle.
Altitude of Triangle – Formulas
The following are the formulas to find the height or altitude of the triangle.
- The general formula is height = (2 x area)/base.
- Equilateral Triangle: h = a√3/2
- Isosceles Triangle: h = √(a² – b²/4)
- Right Triangle: h = √(xy)
- Scalene triangle: h = [2√[(s – a)(s – b)(s – c)]]/b
Frequently Asked Question’s on Altitude Geometrical Property
1. What is the altitude of a triangle?
The altitude of a triangle is the line segment that is drawn the vertex to the opposite side of the triangle. It is the perpendicular line that connects the base and opposite sides.
2. What is the property of altitude?
The altitude is the shortest distance from the vertex to its opposite side. It has 3 altitudes and all of them meet at one point called the ortho-centre of the triangle.
3. What is the difference between altitude and median?
The median is the line segment drawn from the vertex to the opposite side. The altitude is the perpendicular distance from the base to the opposite vertex. Median divides the triangle into two equal parts. Whereas altitudes do not divide the triangle into two equal parts.