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Altitude of a Triangle(Definition & Properties)

altitude of a triangle

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A triangle is one of the most important 2D shapes in geometry. The triangles have certain properties which are used to solve problems. One such property is the altitude of a triangle.

Let’s understand what is the altitude of a triangle and its properties.

What is the Altitude of a Triangle?

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. The altitude is commonly referred to as the height of a triangle and is denoted by the letter $h$.

altitude of a triangle

The length of altitude is measured by calculating the distance between the vertex and its opposite side. You can draw three altitudes in every triangle from each of the vertices to the opposite sides.

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Properties of Altitude of Triangle

The following are the properties of the altitude of a triangle that helps you identify it.

  • 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 $90^{\circ}$ 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

How to Find the Altitude of a Triangle?

There are different types of triangles. For different triangles, there are different formulas for finding the altitude of a triangle. Let’s now discuss these formulas.

Altitude of a Scalene Triangle

In a scalene triangle, there are three altitudes of different lengths. To find the altitude of a scalene triangle, we use Heron’s formula given by $h = \frac{2 \sqrt{s(s – a)(s – b)(s – c)}}{b}$, where $h$ is the height or altitude of the triangle, $s$ is the semi-perimeter of the triangle and  $a$, $b$, and $c$ are the sides of the triangle.

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

Step 1: Write down Heron’s formula for the area of triangle $A = \sqrt{s(s – a)(s – b)(s – c)}$

Step 2: Write down the basic formula for the area of triangle $A =  \frac{1}{2} \times b \times h$

Step 3: Equate the two formulas $\frac{1}{2} \times b \times h = \sqrt{s(s – a)(s – b)(s – c)}$

Step 4: Simplify the expression in Step 3 to get $h = \frac{2 \sqrt{s(s – a)(s – b)(s – c)}}{b}$

Altitude of an Isosceles Triangle

An isosceles triangle is the one whose any of the two sides are equal. In the case of the isosceles triangle also, the altitude of an isosceles triangle is perpendicular to its base.

Let’s derive the formula for the altitude of an isosceles triangle. In the isosceles triangle in the above figure, side $\text{AB} = \text{AC}$, $\text{BC}$ is the base, and $\text{AD}$ is the altitude. 

Let’s represent $\text{AB}$ and $\text{AC}$ as $a$, $\text{BC}$ as $b$, and $\text{AD}$ as $h$. 

We use one of the properties of the altitude of an isosceles triangle that it is the perpendicular bisector to the base of the triangle. 

By applying Pythagoras theorem in $\triangle \text{ADB}$, we get,

$\text{AD}^2 = \text{AB}^2- \text{BD}^2$ _________________ (1)

Since $\text{AD}$ is the bisector of side $\text{BC}$, it divides it into 2 equal parts.

So, $\text{BD} = \frac{1}{2} \times \text{BC}$

Substituting the value of $\text{BD}$ in equation 1, we get

$\text{AD}^2 = \text{AB}^2- \text{BD}^2$

$=>h^2 = a^2 – \left(\frac{1}{2} \times b \right)^2$

$=> h = \sqrt{a^2 – \frac{1}{4}b^2}$

Altitude of an Equilateral Triangle

An equilateral triangle is a triangle in which all three sides are equal. Let’s consider the length of each side of the equilateral triangle as $a$, then its perimeter becomes $3a$ and hence the semi perimeter will be given by $s = \frac{3a}{2}$ and the base of the triangle as $a$.

Now, let’s derive 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 $\frac{3a}{2}$ in this case.

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

$=> h = \frac{2}{a} \sqrt{\frac{3a}{2} \left(\frac{3a}{2} – a \right) \left(\frac{3a}{2} – a \right) \left(\frac{3a}{2} – a \right)}$

$=>h = \frac{2}{a} \sqrt{\frac{3a}{2} \times \frac{a}{2} \times \frac{a}{2} \times \frac{a}{2}}$

$=>h = \frac{2}{a} \times \frac{a^2 \sqrt{3}}{4}$

$=>h = \frac{a \sqrt{3}}{2}$

Altitude of a Right Triangle

A right triangle is a triangle in which one of the angles is $90^{\circ}$. When we construct an altitude of a triangle from a vertex to the hypotenuse of a right-angled triangle, it forms two similar triangles. We use this property of a right triangle to derive the formula for its altitude.

In the above figure, $\triangle \text{PSR} \sim \triangle \text{RSQ}$

Therefore, $\frac{\text{PS}}{\text{RS}} = \frac{\text{RS}}{\text{SQ}}$

$=> \text{RS}^2 = \text{PS} \times \text{SQ}$

This can also be written as: $h^2 = x \times y$, where $x$ and $y$ are the bases of the two similar triangles: $\triangle \text{PSR}$ and $\triangle \text{RSQ}$.

Therefore, the altitude of a right triangle is given by $h = \sqrt{xy}$.

Greatest Math Discoveries

Difference Between Altitude and Median of a Triangle

The following are the differences between the median and altitude of a triangle.

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

Practice Problems

  1. What is an altitude of a triangle?
  2. How many altitudes a triangle can have?
  3. The point of intersection of all the altitudes of a triangle is called ________.
  4. The area of a triangle is $48$ square units. Find the length of the altitude if the length of the base is $6$ units.
  5. Calculate the length of the altitude of a scalene triangle whose sides are $7$ units, $8$ units, and $9$ units respectively.

FAQs

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 that it touches.

How many altitudes a triangle can have?

A triangle can have three altitudes.

What is the point of intersection of all the altitudes of a triangle called?

The point of intersection of three altitudes of a triangle is called its orthocenter.

Conclusion

The altitude of a triangle is the perpendicular line segment drawn from the vertex of the triangle to the side opposite to it. Any triangle can have three altitudes and the point of intersection of these three altitudes of a triangle is called its orthocenter.

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