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A triangle's altitude is the perpendicular traced from the triangle's vertex to the opposite side.
What is the altitude of a triangle?
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Source: SafaltaThe altitude, also known as the triangle's height, forms a right-angle triangle with the base.
Basic Maths Formula
CBSE Class 12 Syllabus 2022-23
2023 CBSE Class 12 Exam Preparation Tips
The basic application of altitude is to calculate the area of a triangle, i.e.
the area of a triangle is (12 base height).
Now, given the area and height of a triangle, the base can be readily determined as Base = [(2 Area)/Height].
What Is the Purpose of a Triangle's Altitude?
The following are the many characteristics of a triangle's altitude:
Triangle's Altitude Properties
- A triangle can have a maximum of three elevations.
- A triangle's altitude is perpendicular to the opposing side. As a result, it makes a 90-degree angle with the opposing side.
- The height might be inside or outside the triangle depending on the kind of triangle.
- The orthocenter of the triangle is the place at which three altitudes intersect.
When it comes to altitude, various triangles have distinct sorts.
The table below provides an overview of several sorts of elevations in various triangles.
Different Triangles' Altitudes
The height of an obtuse-angled triangle is outside the triangle. The base of such triangles is stretched, and a perpendicular is drawn from the opposing vertex to the base.
An equilateral triangle's altitude or height is the line segment from a vertex perpendicular to the opposite side.
It's worth noting that the height of an equilateral triangle cuts across its base and opposing angle.
The diagram below depicts an equilateral triangle ABC in which "BD" represents the height (h), AB = BC = AC, ABD = CBD, and AD = CD.
Equilateral Triangle Altitude
All angles in an equilateral triangle are equal to 60°.
In the ADB triangle,
60° sin = h/AB
We all know that AB = BC = AC = s. (since all sides are equal)
∴ sin 60° = h/s
√3/2 = h/s
h = (√3/2)s
As a result, an equilateral triangle's altitude (height) = h = (3/2) s
A right-angled triangle's altitude separates the current triangle into two comparable triangles.
The geometric mean of line segments created by altitude on the hypotenuse is equal to the altitude on the hypotenuse, according to the right triangle altitude theorem.
When a perpendicular is traced from the vertex to the hypotenuse of a right triangle, two comparable right triangles are created.
This is referred to as the right triangle altitude theorem.
Right Triangle Altitude
The isosceles triangle altitude bisects the vertex angle and the base.
It should be noticed that an isosceles triangle has two congruent sides, hence the altitude cuts across the base and vertex.
Isosceles Triangle Altitude
Triangle Altitudes Formulas
|Triangle Type||Altitude Formula|
|Equilateral Triangle||h = (½) × √3 × s|
|Isosceles Triangle||h =√(a2−b2/4)|
|Right Triangle||h =√(xy)|