how to find altitude of a triangle

The altitude of a triangle is the perpendicular line segment drawn from the vertex to the contrary side of the triangle. It may lie within or outside the triangle, based on the types of triangles. The altitude of a triangle basically defines the height, when nosotros accept to measure out the area of a triangle, with respect to the base.

Table of Contents:

• Definition
• Utilize
• Properties
• Altitude of triangles
  • Obtuse triangle
  • Equilateral triangle
  • Right triangle
  • Isosceles triangle
• Formulas
• Median vs Distance
• Solved examples
• Do 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. Also, known every bit the top of the triangle, the altitude makes a correct-angle triangle with the base. Below is an image that shows a triangle's distance.

What is the Employ of Distance of a Triangle?

The main awarding use of altitude is that information technology is used for area calculation of the triangle, i.e. area of a triangle is (½ base of operations × summit). Now, using the area of a triangle and its height, the base of operations can be hands calculated every bit Base of operations = [(ii × Expanse)/Height]

Properties of Altitude of a Triangle

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

• In that location are a maximum of three altitudes for a triangle
• The altitude of a triangle is perpendicular to the opposite side. Thus, information technology forms 90 degrees angle with the opposite side.
• Depending on the type of triangle, the altitude can prevarication inside or outside the triangle
• The point of intersection of 3 altitudes is called the orthocenter of the triangle

Altitudes of Different Triangles

About distance, dissimilar triangles have different types of altitude. Beneath is an overview of dissimilar types of altitudes in dissimilar triangles.

For an obtuse-angled triangle, the distance is outside the triangle. For such triangles, the base is extended, and so a perpendicular is drawn from the opposite vertex to the base. For an obtuse triangle, the distance is shown in the triangle below.

Altitude of an Obtuse Triangle

Distance of an Equilateral Triangle

The altitude or height of an equilateral triangle is the line segment from a vertex that is perpendicular to the contrary side. It is interesting to notation 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 summit (h), AB = BC = Ac, ∠ABD = ∠CBD, and AD = CD.

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

In triangle ADB,

sin 60° = h/AB

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

∴ sin 60° = h/s

√3/2 = h/due south

h = (√3/2)s

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

Distance of a Correct Triangle

The altitude of a correct-angled triangle divides the existing triangle into two similar triangles. According to theright triangle altitude theorem, the altitude on the hypotenuse is equal to the geometric mean of line segments formed by altitude on the hypotenuse.

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

In the above effigy,

△ADB ∼ △BDC

Thus,

Advert/BD = BD/DC

BD ii = AD.DC

h two = 10.y

h = √xy

Hence, is the altitude of a correct triangle.

Altitude of an Isosceles Triangle

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

Altitudes of a Triangles Formulas

Triangle Type	Distance Formula
Equilateral Triangle	h = (½) × √3 × southward
Isosceles Triangle	h =√(a2−b2⁄ii)
Right 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.	Altitude is drawn from the vertex and is perpendicular to the opposite side of the triangle
It bisects the opposite side	It may or may not bifurcate the contrary side, based on the type of triangle
It lies inside the triangle always	It may or may non lie inside the triangle, depending on the type of triangle
Information technology divides the triangle into 2 equal parts	Information technology does not separate the triangle into two equal parts
The intersection betoken of the iii medians is called the centroid of the triangle	The intersection bespeak of iii altitudes is called the orthocenter of the triangle

Solved Examples

Q.one: 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 = s√3/2

= 4√3/ii

= 2√3 cm

Q.2: If sides of a triangle are a = 3, b = 6, and c = seven, then what is the distance 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, we have;

h = [2√(south(s−a).(s−b).(s−c))]/b

s = (a+b+c)/2 = (3+half-dozen+7)/2 = 16/ii = 8

h = [ii√(8(8-3)(8-6)(8-7))]/two

h = [ii√(viii.5.2.one)]/2

h = iv√v

Practice Questions

1. What is the height of an isosceles triangle, if the length of equal sides is 8 cm and the unequal side is 6 cm?
2. 3 sides of a given triangle are viii units, eleven units, and 13 units. Discover the length of altitude of the triangle.
3. Find the top of an equilateral triangle whose side measures 10 cm.

Related Articles

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

Frequently Asked Questions – FAQs

What is an altitude of a triangle?

An distance 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 distance of a triangle varies for unlike triangles.
For scalene triangle, the altitude is [2√(s(s−a).(s−b).(s−c))]/b
For an equilateral triangle, the altitude is a√3/2
For an isosceles triangle, the altitude is √(a^two – bⁱⁱ/four)
For the correct triangle, the distance is √xy

Where does the distance 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 outside the triangle. It is at 90 degrees angle to the reverse side. The betoken of intersection of three altitudes is called the orthocenter of the triangle.

Is the altitude of an obtuse triangle inside the triangle?

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

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