Altitude of a Triangle

The altitude of a triangle is a perpendicular that is drawn from the vertex of a triangle to the opposite side. Since there are three sides in a triangle, three altitudes can be drawn in it. Different triangles have different kinds of altitudes. The altitude of a triangle which is also called its height is used in calculating the area of a triangle and is denoted by the letter 'h'.

1.	Altitude of a Triangle Definition
2.	Altitude of Triangle Properties
3.	Altitude of Triangle Formula
4.	Difference Between Median and Altitude of Triangle
5.	FAQs on Altitude of a Triangle

Altitude of a Triangle Definition

The altitude of a triangle is the perpendicular line segment drawn from the vertex of the triangle to the side opposite to it. The altitude makes a right angle with the base of the triangle that it touches. It is commonly referred to as the height of a triangle and is denoted by the letter 'h'. It can be measured by calculating the distance between the vertex and its opposite side. It is to be noted that three altitudes can be drawn in every triangle from each of the vertices. Observe the following triangle and see the point where all the three altitudes of the triangle meet. This point is known as the 'Orthocenter'.

Altitude of Triangle

Altitude of a Triangle Properties

The altitudes of various types of triangles have some properties that are specific to certain triangles. They are as follows:

A triangle can have three altitudes.
The altitudes can be inside or outside the triangle, depending on the type of triangle.
The altitude makes an angle of 90° to the side opposite to it.
The point of intersection of the three altitudes of a triangle is called the orthocenter of the triangle.

Altitude of a Triangle Formula

The basic formula to find the area of a triangle is: Area = 1/2 × base × height, where the height represents the altitude. Using this formula, we can derive the formula to calculate the height (altitude) of a triangle: Altitude = (2 × Area)/base. Let us learn how to find out the altitude of a scalene triangle, equilateral triangle, right triangle, and isosceles triangle.

The important formulas for the altitude of a triangle are summed up in the following table. The following section explains these formulas in detail.

Scalene Triangle	\(h= \frac{2 \sqrt{s(s-a)(s-b)(s-c)}}{b}\)
Isosceles Triangle	\(h= \sqrt{a^2- \frac{b^2}{4}}\)
Equilateral Triangle	\(h= \frac{a\sqrt{3}}{2}\)
Right Triangle	\(h= \sqrt{xy}\)

Altitude of a Scalene Triangle

A scalene triangle is one in which all three sides are of different lengths. To find the altitude of a scalene triangle, we use the Heron's formula as shown here. \(h=\dfrac{2\sqrt{s(s-a)(s-b)(s-c)}}{b}\) Here, h = height or altitude of the triangle, 's' is the semi-perimeter; 'a, 'b', and 'c' are the sides of the triangle.

Altitude of a Scalene Triangle

The steps to derive the formula for the altitude of a scalene triangle are as follows:

The area of a triangle using the Heron's formula is, \(Area= \sqrt{s(s-a)(s-b)(s-c)}\).
The basic formula to find the area of a triangle with respect to its base 'b' and altitude 'h' is: Area = 1/2 × b × h
If we place both the area formulas equally, we get, \[\begin{align} \dfrac{1}{2}\times b\times h = \sqrt{s(s-a)(s-b)(s-c)} \end{align}\]
Therefore, the altitude of a scalene triangle is \[\begin{align} h = \dfrac{2\sqrt{s(s-a)(s-b)(s-c)}}{b} \end{align}\]

Altitude of an Isosceles Triangle

A triangle in which two sides are equal is called an isosceles triangle. The altitude of an isosceles triangle is perpendicular to its base.

Let us see the derivation of the formula for the altitude of an isosceles triangle. In the isosceles triangle given above, side AB = AC, BC is the base, and AD is the altitude. Let us represent AB and AC as 'a', BC as 'b', and AD as 'h'. One of the properties of the altitude of an isosceles triangle that it is the perpendicular bisector to the base of the triangle. So, by applying Pythagoras theorem in △ADB, we get,

AD²= AB²- BD² ....(Equation 1)

Since, AD is the bisector of side BC, it divides it into 2 equal parts.

So, BD = 1/2 × BC

Substitute the value of BD in Equation 1,

AD² = AB²- BD²

\(h^2=a^2-(\dfrac{1}{2}\times b)^2\)

\(h=\sqrt{a^2-\dfrac{1}{4}b^2}\)

Altitude of an Equilateral Triangle

A triangle in which all three sides are equal is called an equilateral triangle. Considering the sides of the equilateral triangle to be 'a', its perimeter = 3a. Therefore, its semi-perimeter (s) = 3a/2 and the base of the triangle (b) = a.

altitude of an equilateral triangle

Let us see the derivation of the formula for the altitude of an equilateral triangle. Here, a = side-length of the equilateral triangle; b = the base of an equilateral triangle which is equal to the other sides, so it will be written as 'a' in this case; s = semi perimeter of the triangle, which will be written as 3a/2 in this case.

\(\begin{align} h=\dfrac{2\sqrt{s(s-a)(s-b)(s-c)}}{b} \end{align}\)

\(\begin{align} h=\dfrac{2}{a} \sqrt{\dfrac{3a}{2}(\dfrac{3a}{2}-a)(\dfrac{3a}{2}-a)(\dfrac{3a}{2}-a)} \end{align}\)

\(\begin{align} h=\dfrac{2}{a}\sqrt{\dfrac{3a}{2}\times \dfrac{a}{2}\times \dfrac{a}{2}\times \dfrac{a}{2}} \end{align}\)

\(\begin{align} h=\dfrac{2}{a} \times \dfrac{a^2\sqrt{3}}{4} \end{align}\)

\(\begin{align} \therefore h=\dfrac{a\sqrt{3}}{2} \end{align}\)

Altitude of a Right Triangle

A triangle in which one of the angles is 90° is called a right triangle or a right-angled triangle. When we construct an altitude of a triangle from a vertex to the hypotenuse of a right-angled triangle, it forms two similar triangles. It is popularly known as the Right triangle altitude theorem.

Altitude of a Right Triangle

Let us see the derivation of the formula for the altitude of a right triangle. In the above figure, △PSR ∼ △RSQ

So, \(\dfrac{PS}{RS}=\dfrac{RS}{SQ}\)

RS²= PS × SQ

Referring to the figure given above, this can also be written as: h² = x × y, here, 'x' and 'y' are the bases of the two similar triangles: △PSR and △RSQ.

Therefore, the altitude of a right triangle (h) = √xy

Altitude of an Obtuse Triangle

A triangle in which one of the interior angles is greater than 90° is called an obtuse triangle. The altitude of an obtuse triangle lies outside the triangle. It is usually drawn by extending the base of the obtuse triangle as shown in the figure given below.

Altitude of an Obtuse Triangle

Difference Between Median and Altitude of Triangle

We know that the median and the altitude of a triangle are line segments that join the vertex to the opposite side of a triangle. However, they are different from each other in many ways. Observe the figure and the table given below to understand the difference between the median and altitude of a triangle.

Difference between median and altitude of triangle

Median of a Triangle	Altitude of a Triangle
The median of a triangle is the line segment drawn from the vertex to the opposite side.	The altitude of a triangle is the perpendicular distance from the base to the opposite vertex.
It always lies inside the triangle.	It can be both outside or inside the triangle depending on the type of triangle.
It divides a triangle into two equal parts.	It does not divide the triangle into two equal parts.
It bisects the base of the triangle into two equal parts.	It does not bisect the base of the triangle.
The point where the 3 medians of a triangle meet is known as the centroid of the triangle.	The point where the 3 altitudes of the triangle meet is known as the orthocenter of that triangle.

Important Notes

Here is a list of a few important points related to the altitude of a triangle.

The point where all the three altitudes of a triangle intersect is called the orthocenter.
Both the altitude and the orthocenter can lie inside or outside the triangle.
In an equilateral triangle, the altitude is the same as the median of the triangle.

Topics Related to Altitude of a Triangle

Check out some interesting topics related to the altitude of a triangle.

Solved Examples on Altitude of a Triangle

Example 1: The area of a triangle is 72 square units. Find the length of the altitude if the length of the base is 9 units.

Solution:

We know that altitude of a triangle, h = (2 × Area) / Base.
Given, area = 72 square units and base = 9 units.
Altitude 'h' = (2 × 72) / 9
= 144/9
= 16 units.
Therefore, altitude 'h' = 16 units.
Example 2: Calculate the length of the altitude of a scalene triangle whose sides are 7 units, 8 units, and 9 units respectively.

Solution:

The perimeter of a triangle is the sum of all the sides = 7 + 8 + 9 = 24 units. Semi-perimeter (s) = 24/2 =12 units. Let us name the sides of the scalene triangle to be 'a', 'b', and 'c' respectively. Therefore, a = 9 units, b = 8 units and c = 7 units;

The altitude of the triangle:

\(h= \frac{2 \sqrt{s(s-a)(s-b)(s-c)}}{b}\)

Altitude(h) = \(\frac{2 \sqrt{12(12-9)(12-8)(12-7)}}{8}\)

Altitude(h) = \(\frac{2 \sqrt{12\ \times 3\ \times 4\ \times 5}}{8}\)

Altitude (h) = 6.70 units
Example 3: Calculate the altitude of an isosceles triangle whose two equal sides are 8 units and the third side is 6 units.

Solution:

The equal sides (a) = 8 units, the third side (b) = 6 units. In an isosceles triangle the altitude is:

\(h= \sqrt{a^2- \frac{b^2}{4}}\)

Altitude(h)= \(\sqrt{8^2-\frac{6^2}{4}}\)

Altitude(h)= √[64-(36/4)]

Altitude(h)= √55

Altitude(h)= 7.41 units

Therefore, the altitude of the isosceles triangle is 7.41 units.

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Practice Questions on Altitude of a Triangle

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FAQs on Altitude of a Triangle

What is the Altitude of a Triangle in Math?

The altitude of a triangle is a line segment that is drawn from the vertex of a triangle to the side opposite to it. It is perpendicular to the base or the opposite side which it touches. Since there are three sides in a triangle, three altitudes can be drawn in a triangle. All the three altitudes of a triangle intersect at a point called the 'Orthocenter'.

What is the Altitude of Triangle Formula?

The altitude of a triangle can be calculated according to the different formulas defined for the various types of triangles. The formulas used for the different triangles are given below:

Altitude of a scalene triangle = \(h= \frac{2 \sqrt{s(s-a)(s-b)(s-c)}}{b}\); where 'a', 'b', 'c' are the 3 sides of the triangle; 's' is the semi perimeter of the triangle.
Altitude of an isosceles triangle = \(h= \sqrt{a^2- \frac{b^2}{4}}\); where 'a' is one of the equal sides, 'b' is the third side of the triangle.
Altitude of an equilateral triangle = \(h= \frac{a\sqrt{3}}{2}\); where 'a' is one side of the triangle
Altitude of a right triangle = \(h= \sqrt{xy}\); where 'x' and 'y' are the bases of the two similar triangles formed.

What are the Properties of Altitude of a Triangle?

The altitude of a triangle is the line drawn from a vertex to the opposite side of a triangle. The important properties of the altitude of a triangle are as follows:

A triangle can have three altitudes.
The altitudes can be inside or outside the triangle, depending on the type of triangle.
The altitude makes an angle of 90° to the side opposite to it.
The point of intersection of the three altitudes of a triangle is called the orthocenter of the triangle.

How to Find the Altitude of a Right Triangle?

A triangle in which one of the angles is 90° is a right triangle. When an altitude is drawn from a vertex to the hypotenuse of a right-angled triangle, it forms two similar triangles. The formula to calculate the altitude of a right triangle is h =√xy. where 'h' is the altitude of the right triangle and 'x' and 'y' are the bases of the two similar triangles formed after drawing the altitude from a vertex to the hypotenuse of the right triangle.

How to Find the Altitude of a Scalene Triangle?

A triangle in which all three sides are unequal is a scalene triangle. The formula to calculate the altitude of a scalene triangle is \(h= \frac{2 \sqrt{s(s-a)(s-b)(s-c)}}{b}\), where 'h' is the altitude of the scalene triangle; 's' is the semi-perimeter, which is half of the value of the perimeter, and 'a', 'b' and 'c' are three sides of the scalene triangle.

What is the Difference Between Median and Altitude of Triangle?

The altitude of a triangle and median are two different line segments drawn in a triangle. The altitude of a triangle is the perpendicular distance from the base to the opposite vertex. It can be located either outside or inside the triangle depending on the type of triangle. The median of a triangle is the line segment drawn from the vertex to the opposite side that divides a triangle into two equal parts. It bisects the base of the triangle and always lies inside the triangle.

Does the Altitude of a Triangle Always Make 90° With the Base of the Triangle?

Yes, the altitude of a triangle is a perpendicular line segment drawn from a vertex of a triangle to the base or the side opposite to the vertex. Since it is perpendicular to the base of the triangle, it always makes a 90° with the base of the triangle.

Is the Altitude of a Triangle Same as the Height of a Triangle?

Yes, the altitude of a triangle is also referred to as the height of the triangle. It is denoted by the small letter 'h' and is used to calculate the area of a triangle. The formula for the area of a triangle is (1/2) × base × height. Here, the 'height' is the altitude of the triangle.

Does the Altitude of an Obtuse Triangle lie Inside the Triangle?

No, the altitude of an obtuse triangle lies outside the triangle because the angle opposite to the vertex from which the altitude is drawn is an obtuse angle. This is done by extending the base of the given obtuse triangle.

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