How To Find The Altitude Of A Right Triangle

The altitude of a triangle is the perpendicular line segment drawn from the vertex to the opposite side of the triangle. It may lie within or outside the triangle, based on the types of triangles. The distance of a triangle basically defines the pinnacle, when we take to mensurate the area of a triangle, with respect to the base of operations.

Table of Contents:

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

What is Altitude Of A Triangle?

The altitude of a triangle is the perpendicular fatigued from the vertex of the triangle to the opposite side. Likewise, known as the height of the triangle, the distance makes a right-angle triangle with the base. Beneath is an paradigm that shows a triangle'southward altitude.

What is the Use of Altitude of a Triangle?

The primary awarding use of altitude is that information technology is used for surface area calculation of the triangle, i.east. expanse of a triangle is (½ base of operations × superlative). At present, using the area of a triangle and its peak, the base can be easily calculated as Base = [(2 × Area)/Height]

Properties of Distance of a Triangle

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

• There are a maximum of iii altitudes for a triangle
• The distance of a triangle is perpendicular to the reverse side. Thus, it forms ninety degrees angle with the opposite side.
• Depending on the blazon of triangle, the altitude can lie inside or exterior the triangle
• The bespeak of intersection of three altitudes is chosen the orthocenter of the triangle

Altitudes of Dissimilar Triangles

About distance, dissimilar triangles have different types of altitude. Beneath 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 fatigued from the opposite vertex to the base. For an obtuse triangle, the altitude is shown in the triangle beneath.

Altitude of an Birdbrained Triangle

Altitude of an Equilateral Triangle

The altitude or height of an equilateral triangle is the line segment from a vertex that is perpendicular to the opposite side. It is interesting to note that the altitude of an equilateral triangle bisects its base of operations and the reverse bending. The paradigm below shows an equilateral triangle ABC where "BD" is the height (h), AB = BC = Air conditioning, ∠ABD = ∠CBD, and AD = CD.

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

In triangle ADB,

sin 60° = h/AB

We know, AB = BC = AC = south (since all sides are equal)

∴ sin threescore° = h/s

√three/2 = h/s

h = (√3/ii)southward

Therefore, the Altitude (Height) of an equilateral triangle = h = √(3⁄2) × southward

Distance of a Right Triangle

The altitude of a right-angled triangle divides the existing triangle into 2 like triangles. According to thecorrect triangle altitude theorem, the altitude on the hypotenuse is equal to the geometric mean of line segments formed past distance on the hypotenuse.

For a right triangle, when a perpendicular is drawn from the vertex to the hypotenuse, ii like right triangles are formed. This is called the right triangle altitude theorem.

In the above effigy,

△ADB ∼ △BDC

Thus,

AD/BD = BD/DC

BD 2 = 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 bending of the vertex and bisects the base. It should be noted that an isosceles triangle is a triangle with two coinciding sides and and so, the altitude bisects the base and vertex.

Altitudes of a Triangles Formulas

Triangle Type	Altitude Formula
Equilateral Triangle	h = (½) × √3 × southward
Isosceles Triangle	h =√(a2−b2⁄2)
Right Triangle	h =√(xy)

Difference 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.	Altitude is drawn from the vertex and is perpendicular to the reverse side of the triangle
It bisects the reverse side	It may or may not bisect the reverse side, based on the type of triangle
It lies within the triangle always	It may or may not lie inside the triangle, depending on the type of triangle
It divides the triangle into 2 equal parts	It does not separate the triangle into two equal parts
The intersection bespeak of the three medians is called the centroid of the triangle	The intersection betoken of iii altitudes is called the orthocenter of the triangle

Solved Examples

Q.1: What is the distance of an equilateral triangle, if its side length is equal to 4 cm?

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

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

= 4√3/2

= 2√3 cm

Q.2: If sides of a triangle are a = three, b = 6, and c = 7, and so 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 altitude of scalene triangle, nosotros have;

h = [2√(s(s−a).(due south−b).(s−c))]/b

southward = (a+b+c)/2 = (3+6+seven)/2 = sixteen/two = 8

h = [2√(8(8-3)(eight-6)(8-seven))]/two

h = [2√(viii.v.2.1)]/2

h = 4√five

Practice Questions

1. What is the peak of an isosceles triangle, if the length of equal sides is eight cm and the unequal side is half dozen cm?
2. Iii sides of a given triangle are 8 units, xi units, and xiii units. Find the length of distance of the triangle.
3. Find the elevation of an equilateral triangle whose side measures 10 cm.

Related Articles

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

Frequently Asked Questions – FAQs

What is an distance of a triangle?

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

What is the formula for an distance of a triangle?

The formula for an distance of a triangle varies for different triangles.
For scalene triangle, the altitude is [2√(due south(south−a).(s−b).(s−c))]/b
For an equilateral triangle, the altitude is a√3/2
For an isosceles triangle, the altitude is √(aⁱⁱ – b²/4)
For the right triangle, the altitude is √xy

Where does the altitude of an acute triangle lie?

The altitude of an astute triangle lies inside the triangle.

What is the belongings of the altitude of a triangle?

The altitude of a triangle lies inside or exterior the triangle. It is at 90 degrees angle to the opposite side. The bespeak of intersection of three altitudes is called the orthocenter of the triangle.

Is the altitude of an birdbrained triangle inside the triangle?

No, the distance of the birdbrained triangle lies outside the triangle.

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