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'.
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 it touches. It is commonly referred to as the height of a triangle and is denoted by the letter 'h'. There are three altitudes in every triangle drawn from each of the vertices. The altitude of a triangle can be measured by calculating the distance between the vertex and its opposite side.
Observe the following triangle and see the point where all the three altitudes meet inside a triangle. This point is known as the 'Orthocenter'. In a right-angled triangle, the altitude of a triangle is used in trigonometric calculations.
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.
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.
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 general formula to find the area of a triangle with respect to its base 'b' and altitude 'h' is Area = 1/2 × b × h
- Equating area obtained from both the formulas, 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 \[\begin{align} h = \dfrac{2\sqrt{s(s-a)(s-b)(s-c)}}{b} \end{align}\].
The area of a triangle using the Heron's formula is:
\(Area= \sqrt{s(s-a)(s-b)(s-c)}\)
The general formula to find the area of a triangle with respect to its base 'b' and altitude 'h' is,
Area = 1/2 × b × h
Placing both the equations equally, we get:
\[\begin{align} \dfrac{1}{2}\times b\times h=\sqrt{s(s-a)(s-b)(s-c)} \end{align}\]
\[\begin{align} h=\dfrac{2\sqrt{s(s-a)(s-b)(s-c)}}{b} \end{align}\]
This is how we got our formula to find the altitude of a scalene triangle.
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.
Let us see the derivation of the formula for the altitude of an equilateral triangle. Here, a = side-length of the equilateral triangle; b= base of the triangle (which is equal to the common side-length in case of an equilateral triangle)
\(\begin{align} h=\dfrac{2\sqrt{s(s-a)(s-b)(s-c)}}{b} \end{align}\)
\(\begin{align} h=\dfrac{2}{a} \sqrt{\dfrac{3a}{2}(\dfrac{3a}{2}-a)(\dfrac{3a}{2}-a)(\dfrac{3a}{2}-a)} \end{align}\)
\(\begin{align} h=\dfrac{2}{a}\sqrt{\dfrac{3a}{2}\times \dfrac{a}{2}\times \dfrac{a}{2}\times \dfrac{a}{2}} \end{align}\)
\(\begin{align} h=\dfrac{2}{a} \times \dfrac{a^2\sqrt{3}}{4} \end{align}\)
\(\begin{align} \therefore h=\dfrac{a\sqrt{3}}{2} \end{align}\)
Altitude of a Right Triangle
A triangle in which one of the angles is a right angle (or a 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.
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
h2 = x × y, here, 'x' and 'y' are the bases of the two similar triangles triangle PSR and triangle RSQ.
Therefore, h = √xy
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 given isosceles triangle, side AB and side AC are equal, 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
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 the above equation,
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 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 Triangles Formulas
The important formulas for the altitude of a triangle are summed up in the following table.
| Scalene Triangle | \(h= \frac{2 \sqrt{s(s-a)(s-b)(s-c)}}{b}\) |
|---|---|
| Equilateral Triangle | \(h= \frac{a\sqrt{3}}{2}\) |
| Right Triangle | \(h= \sqrt{xy}\) |
| Isosceles Triangle | \(h= \sqrt{a^2- \frac{b^2}{4}}\) |
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 and Median of a Triangle
The altitude of a triangle is the perpendicular distance from the base to the opposite vertex. It can be both outsides 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 angle formed at the vertex from where it is drawn and the base of the triangle. It always lies inside the triangle.
Important Notes
A few important points to remember about the altitude of a triangle are given below.
- The point where all the three altitudes in 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
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FAQs on Altitude of a Triangle
What is the Altitude of a Triangle?
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'.
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 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.
What is the Formula to Calculate 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 Formula to Calculate 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 an equilateral 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.
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.
Is the Altitude of a Triangle Same as the Median of the Triangle?
No, 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 both 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 angle formed at the vertex from where it is drawn and also the base of the triangle. It always lies inside the triangle.
Does the Altitude of an Obtuse Triangle lie Inside the Triangle?
No, the altitude of an obtuse triangle lies outside the triangle. It 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.