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What is the altitude of a triangle?
A triangle's altitude is the perpendicular traced from the triangle's vertex to the opposite side. The altitude, also known as the triangle's height, forms a right-angle triangle with the base.
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What Is the Purpose of a Triangle's Altitude?
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].
Triangle's Altitude Properties
The following are the many characteristics of a triangle's altitude:
- 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.
Different Triangles' Altitudes
When it comes to altitude, various triangles have distinct sorts. The table below provides an overview of several sorts of elevations in various triangles.
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.
Equilateral Triangle Altitude
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.
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
Right Triangle Altitude
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.
Isosceles 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.
Triangle Altitudes Formulas
| Triangle Type | Altitude Formula |
|---|---|
| Equilateral Triangle | h = (½) × √3 × s |
| Isosceles Triangle | h =√(a2−b2/4) |
| Right Triangle | h =√(xy) |
The right triangle altitude theorem, also known as the geometric mean theorem, is a basic geometry conclusion that defines a relationship between the hypotenuse altitude in a right triangle and the two line segments it forms on the hypotenuse. The geometric mean of the two segments equals the altitude, according to the formula.