Altitude Of A Triangle Formula

The altitude of a triangle is the perpendicular line segment drawn from the vertex to the opposite side of the triangle. Information technology may prevarication inside or exterior the triangle, based on the types of triangles. The altitude of a triangle basically defines the summit, when we have to measure the area of a triangle, with respect to the base.

Table of Contents:

Definition
Apply
Properties
Altitude of triangles
- Obtuse triangle
- Equilateral triangle
- Right triangle
- Isosceles triangle
Formulas
Median vs Altitude
Solved examples
Practice problems
FAQs

What is Distance Of A Triangle?

The altitude of a triangle is the perpendicular drawn from the vertex of the triangle to the reverse side. As well, known as the height of the triangle, the altitude makes a right-bending triangle with the base. Below is an image that shows a triangle's altitude.

Altitude of a triangle

What is the Use of Altitude of a Triangle?

The main application employ of distance is that it is used for area calculation of the triangle, i.e. area of a triangle is (½ base × elevation). Now, using the area of a triangle and its height, the base of operations can exist easily calculated as Base = [(two × Area)/Height]

Backdrop of Altitude of a Triangle

The different backdrop of altitude of a triangle are listed below:

There are a maximum of three altitudes for a triangle
The altitude of a triangle is perpendicular to the opposite side. Thus, information technology forms xc degrees angle with the opposite side.
Depending on the type of triangle, the altitude can prevarication inside or outside the triangle
The signal of intersection of three altitudes is called the orthocenter of the triangle

Altitudes of Unlike Triangles

Nearly altitude, unlike triangles have unlike types of distance. Below is an overview of different types of altitudes in different triangles.

For an obtuse-angled triangle, the altitude is outside the triangle. For such triangles, the base is extended, and then a perpendicular is drawn from the contrary vertex to the base. For an birdbrained triangle, the altitude is shown in the triangle below.

Altitude of an Obtuse Triangle

Altitude of an Equilateral Triangle

The altitude or tiptop of an equilateral triangle is the line segment from a vertex that is perpendicular to the contrary side. It is interesting to annotation that the altitude of an equilateral triangle bisects its base and the opposite angle. The image below shows an equilateral triangle ABC where "BD" is the height (h), AB = BC = AC, ∠ABD = ∠CBD, and AD = CD.

Altitude of an Equilateral Triangle

For an equilateral triangle, all angles are equal to 60°.

In triangle ADB,

sin 60° = h/AB

We know, AB = BC = Ac = s (since all sides are equal)

∴ sin 60° = h/southward

√3/ii = h/southward

h = (√three/2)s

Therefore, the Distance (Height) of an equilateral triangle = h = (√3/2) × south

Altitude of a Right Triangle

The distance of a right-angled triangle divides the existing triangle into ii similar triangles. According to thecorrect triangle distance theorem, the altitude on the hypotenuse is equal to the geometric mean of line segments formed past distance on the hypotenuse.

Altitude of a Right Triangle

For a right triangle, when a perpendicular is drawn from the vertex to the hypotenuse, two similar right triangles are formed. This is called the right triangle distance theorem.

Altitude of a right angle triangle

In the above effigy,

△ADB ∼ △BDC

Thus,

AD/BD = BD/DC

BD two = AD.DC

h 2 = x.y

h = √xy

Hence, is the altitude of a right triangle.

Altitude of an Isosceles Triangle

The isosceles triangle altitude bisects the angle of the vertex and bisects the base. It should be noted that an isosceles triangle is a triangle with two coinciding sides so, the distance bisects the base and vertex.

Altitude of an Isosceles Triangle

Altitudes of a Triangles Formulas

Triangle Type	Altitude Formula
Equilateral Triangle	h = (½) × √3 × southward
Isosceles Triangle	h =√(aⁱⁱ−b²/4)
Correct Triangle	h =√(xy)

Deviation Between Median and Altitude of a Triangle

Median of triangle	Altitude of triangle
Median is a line segment drawn from the vertex to the middle of the opposite side of a triangle.	Distance is drawn from the vertex and is perpendicular to the contrary side of the triangle
It bisects the opposite side	It may or may not bisect the opposite side, based on the type of triangle
It lies inside the triangle e'er	It may or may non lie within the triangle, depending on the type of triangle
Information technology divides the triangle into ii equal parts	It does not divide the triangle into two equal parts
The intersection point of the 3 medians is called the centroid of the triangle	The intersection betoken of three altitudes is called the orthocenter of the triangle

Solved Examples

Q.1: What is the altitude of an equilateral triangle, if its side length is equal to four cm?

Solution: Given, the side length of an equilateral triangle is 4 cm.

The altitude of an equilateral triangle, h = southward√iii/2

= iv√three/ii

= ii√iii cm

Q.ii: If sides of a triangle are a = three, b = 6, and c = vii, then what is the altitude of the triangle?

Solution: Since all the sides of the given triangle are unequal in length, thus it is a scalene triangle.

Using the formula for an altitude of a scalene triangle, we accept;

h = [2√(s(s−a).(s−b).(south−c))]/b

s = (a+b+c)/2 = (3+six+7)/2 = 16/2 = 8

h = [2√(eight(8-3)(viii-vi)(eight-7))]/2

h = [2√(8.5.2.1)]/2

h = 4√5

Practice Questions

What is the meridian of an isosceles triangle, if the length of equal sides is eight cm and the diff side is half-dozen cm?
Three sides of a given triangle are 8 units, 11 units, and 13 units. Observe the length of altitude of the triangle.
Find the height of an equilateral triangle whose side measures 10 cm.

Triangles
Area of Triangle
Altitude And Median Of A Triangle
Isosceles Triangle
Right Angled Triangle
Equilateral Triangle

Oftentimes Asked Questions – FAQs

What is an altitude of a triangle?

An altitude of a triangle is the perpendicular distance drawn from the vertex to the opposite side of the triangle.

What is the formula for an altitude of a triangle?

The formula for an altitude of a triangle varies for different triangles.
For scalene triangle, the altitude is [2√(due south(s−a).(s−b).(s−c))]/b
For an equilateral triangle, the altitude is a√iii/ii
For an isosceles triangle, the distance is √(a^two – b²/4)
For the right triangle, the altitude is √xy

Where does the altitude of an acute triangle lie?

The distance of an acute triangle lies within the triangle.

What is the property of the distance of a triangle?

The distance of a triangle lies inside or outside the triangle. It is at xc degrees angle to the opposite side. The point of intersection of three altitudes is called the orthocenter of the triangle.

Is the altitude of an obtuse triangle within the triangle?

No, the altitude of the obtuse triangle lies outside the triangle.

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