Altitude of a Triangle

Altitude of a Triangle
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A perpendicular which is drawn from the vertex of a triangle to the opposite side is called the altitude of a triangle.

Try the following simulation and notice the changes in the triangle when you drag the vertices. Move the slider to observe the change in the altitude of the triangle.

Wasn't it interesting? Let's explore the altitude of a triangle in this lesson.

We will learn about the altitude of a triangle, including its definition, altitudes in different types of triangles, formulae, some solved examples and a few interactive questions for you to test your understanding.

Lesson Plan

What is the Altitude of a Triangle?

Observe the picture of the Eiffel Tower given below.

Can you find its height?

Height of Eiffel tower

The height of the Eiffel Tower can also be called its altitude. It is the distance from the base to the vertex of the 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.

It is also known as the height or the perpendicular of the triangle.

There are three altitudes in every triangle drawn from each of the vertex.

The point where all the three altitudes meet inside a triangle is known as the Orthocenter.

Altitude (triangle)


Altitude of a Triangle Formula

We know that the formula to find the area of a triangle is \(\dfrac{1}{2}\times \text{base}\times \text{height}\), where the height represents the altitude.

 So, we can calculate the height (altitude) of a triangle by using this formula:

\(h=\dfrac{2 \times \text{Area}}{\text{base}}\)

Scalene Triangle

A scalene triangle XYZ has all sides unequal.

To find the altitude of a scalene triangle, we use the Heron's formula as shown here.

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

where, h = height or altitude of the triangle;

           s = semi-perimeter;

           a, b and c = the sides of the triangle

Let's understand why we use this formula by learning about its derivation.

Click here to see the proof of derivation

Derivation

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,

\(\text{Area}=\dfrac{1}{2}\times b\times h\)

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 out the altitude of a scalene triangle.

Equilateral Triangle

Equilateral triangle ABC has all sides equal

In case of an equilateral triangle, all the three sides of the triangle are equal.

So, its semi-perimeter is \(s=\dfrac{3a}{2}\) and \(b=a\)

where, a= side-length of the equilateral triangle

            b= base of the triangle (which is equal to the common side-length in case of equilateral triangle)

Let's derive the formula to be used in an equilateral triangle.

Click here to see the proof of derivation.

Derivation

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

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

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

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

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

\(h=\dfrac{a\sqrt{3}}{2}\)

Right 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.

How to measure the altitude of a right triangle - Quora

\(h=\sqrt{xy}\)

Click here to see the proof of derivation and it will open as you click.

Derivation

In the above figure, \(\triangle PSR \sim \triangle RSQ\)

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

      \(RS^2=PS \times SQ\)

      \(h^2=x \times y\)

\(\therefore h=\sqrt{xy}\)

Isosceles Triangle

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 \(h\).

Let's see how to find the altitude of an isosceles triangle with respect to its sides.

Click here to see the proof of derivation and it will open as you click.

Derivation

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 \(\triangle ADB\), we get,

\(AD^2=AB^2-BD^2\)

Since, \(AD\) is the bisector of side \(BC\), it divides it into 2 equal parts, as you can see in the above image.

So, \(BD=\dfrac{1}{2} \times BC(b)\)

Substitute the value of \(BD\) in the above equation,

\(AD^2=AB^2-BD^2\)

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

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

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

Observe the table to go through the formulas used to calculate the altitude (height) of different triangles.

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

The altitude of a Triangle Calculator

Try your hands at the simulation given below. Write the values of base and area and click on 'Calculate' to find the length of altitude.


How do you find the Altitude of a Triangle?

To identify the altitudes in a triangle, we need to identify the type of the triangle.

After identifying the type, we can use the formulas given above to find the value of the altitudes.

Let's visualize the altitude of construction in different types of triangles.

Equilateral Triangle

In an equilateral triangle, altitude of a triangle theorem states that altitude bisects the base as well as the angle at the vertex through which it is drawn. It is the same as the median of the triangle.

altitude in an equilateral triangle

Obtuse Triangle

In an obtuse triangle, the altitude lies outside the triangle. The base is extended and the altitude is drawn from the opposite vertex to this base.

Altitude of an obtuse triangle

Right Triangle

In a right triangle, the altitude from the vertex to the hypotenuse divides the triangle into two similar triangles.

Pythagoras theorem and similar triangles


What Is the 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 outside or inside the triangle depending on the type of the 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.

altitude and median of a triangle

 
important notes to remember
Important Notes
  1. The point where all the three altitudes in a triangle intersect is called the Orthocenter.
  2. Both the altitude and the orthocenter can lie inside or outside the triangle.
  3. In an equilateral triangle, the altitude is the same as the median of the triangle.

Solved Examples

Example 1

 

 

The area of a right triangular swimming pool is 72 sq. units. Find the length of the altitude if the length of the base is 9 units.

Altitude of a triangular swimming pool

Solution

We know that, Altitude of a Triangle, \(h= \frac{2\times\ Area}{base}\)

\(Altitude(h)= \frac{2\ \times\ 72}{9}\)

\(Altitude(h)= 16\ units\)

\(\therefore\) The altitude of the park is 16 units.
Example 2

 

 

Calculate the length of the altitude of the given triangle drawn from the vertex A.

An obtuse scalene triangle

Solution

Perimeter of the triangle is the sum of all the sides, i.e., 24 feet.

Semi-perimeter= 12 feet

Altitude of the Triangle: 

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

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

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

\(Altitude(h)= \frac{ 12\ \sqrt{5}}{4}\)

\(Altitude(h)= 3\sqrt{5}\ feet\)

 

\(\therefore\) The altitude of the given triangle is \(3\sqrt{5} feet\).
Example 3

 

 

Observe the picture of the ladder and find the shortest distance or altitude from the top of the staircase to the ground. 

altitude of a staircase

Solution

In the Staircase, both the legs are of same length, so it forms an isosceles triangle.

In an isosceles triangle the altitude is:

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

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

\(Altitude(h)= \sqrt{64-\frac{36}{2}}\)

\(Altitude(h)= \sqrt{64-18}\)

\(Altitude(h)= \sqrt{46}\ units\)

\(\therefore\) The altitude of the staircase is \(\sqrt{46}\ units\).
 
Challenge your math skills
Challenging Questions
  1. The perimeter of an isosceles triangle is 100 ft. If the base is 36 ft, find the length of the altitude from the vertex formed between the equal sides to the base.
  2.  Find the altitude of a scalene triangle whose two sides are given as 4 units and 7 units, and the perimeter is 19 units.

Interactive Questions on Altitude of a Triangle

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of altitude of a triangle. The math journey around altitude of a triangle starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions(FAQs)

1. What is the altitude of an obtuse triangle?

In an obtuse triangle, the altitude drawn from the obtuse-angled vertex lies interior to the opposite side, while the altitude drawn from the acute-angled vertices lies outside the triangle to the extended opposite side.

2. What is the altitude of an isosceles triangle?

In an isosceles triangle, the altitude drawn from the vertex between the same sides bisects the incongruent side and the angle at the vertex from where it is drawn.

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

3. What are the different types of triangles?

There are many different types of triangles such as the scalene triangle, isosceles triangle, equilateral triangle, right-angled triangle, obtuse-angled triangle and acute-angled triangle.

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