Altitude of an Triangle: Definition Explained with Examples

Triangles have some special line segments like perpendicular bisectors, medians, and altitudes. When you think of altitude, you may picture increasing elevations of mountain ranges. The term also refers to a concept in geometry - it's the height of a triangle.

In this article, we'll explore the concept of altitudes in triangles and related terms in more detail. We'll learn how to calculate the altitude for different types of triangles.

What Is an Altitude?

An altitude is a perpendicular segment drawn from a vertex to the opposite side or extension of the opposite side. In other words, it's a line segment that drops down vertically from a vertex to the base of the triangle.

The altitude also represents the triangle's height. Every triangle has three altitudes, which can lie inside, outside, or on the triangle. Take a look at some examples of altitude positions.

Altitude positions in triangles, ck12.org

Properties of Altitudes

Here are some key properties of altitudes:

• An altitude forms a 90° angle with the side opposite the vertex.
• The location of the altitude varies depending on the triangle type.
• Since a triangle has three vertices, it also has three altitudes.
• The point where the three altitudes intersect is called the orthocenter of the triangle.

Altitude formula for various triangles

There exist different altitude formulas depending on the type of triangle. We will explore the altitude formula for triangles in general as well as specifically for scalene triangles, , , and , along with brief discussions on how these formulas are derived.

General altitude formula

Since altitude is used to determine the area of a triangle, we can derive the formula from the area itself.

Area of a triangle$=\frac{1}{2}\times b\times h$, where b represents the base of the triangle and h is the height/altitude. Therefore, we can infer the height of a triangle as follows:

$$\begin{gathered} Area=\frac{1}{2}\times b\times h \\ \Rightarrow 2\times Area=b\times h \\ \Rightarrow \frac{2\times Area}{b}=h \end{gathered}$$

Altitude (h)$\mathbf{=}\mathbf{(}\mathbf{2}\mathbf{\times }\mathbf{A}\mathbf{r}\mathbf{e}\mathbf{a}\mathbf{)}\mathbf{/}\mathit{b}$

For a triangle$∆ABC$, the area is$81c{m}^2$with a base length of$9cm$. Find the altitude length for this triangle.

Solution: Given the area and base for triangle$∆ABC$, we can directly apply the general formula to find the length of the altitude.

Altitude h$=\frac{2\times Area}{base}=\hspace{0.17em}\frac{2\times 81}{9}=18cm$.

Altitude formula for a scalene triangle

A scalene triangle, which has different side lengths for all three sides, utilizes Heron's formula to derive the altitude.

Heron's formula calculates the triangle's area using the side lengths, perimeter, and semi-perimeter.

Area of a triangle (by Heron's formula) = √(s(s-x)(s-y)(s-z))

Here, s represents the semi-perimeter of the triangle (i.e., s = (x+y+z)/2) and x, y, z are the lengths of sides.

By using the general area formula and equating it with Heron's formula, we can derive the altitude as follows:

Area = 1/2 * b * h

∴ h = 2*((s(s-x)(s-y)(s-z))^(1/2))/b

Thus, the altitude for a scalene triangle is given by the formula: h = 2*((s(s-x)(s-y)(s-z))^(1/2))/b.

In a scalene triangle $∆ABC$, AD is the altitude with base BC. The lengths of all three sides AB, BC, and AC are 12, 16, and 20, respectively. The perimeter for this triangle is given as 48 cm. To calculate the length of the altitude AD, we can follow these steps:

Solution: Given $x=12cm,y=16cm,z=20cm$. Base BC has a length of 16 cm. To calculate the length of the altitude, we first find the semiperimeter:

Semiperimeter $s=\hspace{0.17em}\frac{perimeter}{2}=\frac{48}{2}=24cm.$

Using the formula for the altitude of a scalene triangle, we get:

Altitude $h=\frac{2\left(\sqrt{\mathrm{s}(\mathrm{s}-\mathrm{x})(\mathrm{s}-\mathrm{y})(\mathrm{s}-\mathrm{z})}\right)}{\mathrm{b}}$

Substituting the values, we get:

$$\begin{gathered} =\frac{2\sqrt{24(24-12)(24-16)(24-20)}}{16} \\ =\frac{2\times 96}{16}=12 \end{gathered}$$

Therefore, the length of the altitude AD for this scalene triangle is 12 cm.

Altitude formula for an isosceles triangle

An isosceles triangle has two sides that are equal in length. The altitude of this triangle is the line that is perpendicular to the base and bisects it. The altitude of an isosceles triangle can be found by applying the triangle's properties and Pythagoras' theorem.

In triangle ∆ABC, assuming AB = AC = x, we apply the property of an isosceles triangle that the altitude divides the base side into two equal segments.

⇒ 1/2 BC = DC = BD

When applying Pythagoras' theorem to $∆ABD$, we obtain:

$$\begin{gathered} A{B}^2=A{D}^2+B{D}^2 \\ \Rightarrow A{B}^2=A{D}^2+{\left(\frac{1}{2}BC\right)}^2 \\ \Rightarrow A{D}^2=\hspace{0.17em}A{B}^2-{\left(\frac{1}{2}BC\right)}^2 \end{gathered}$$

By substituting the given side values, we get:

$$\begin{gathered} \Rightarrow h^2=x^2-\frac{1}{4}{y}^2 \\ \therefore h=\sqrt{{\mathrm{x}}^2-\frac{1}{4}{\mathrm{y}}^2} \end{gathered}$$

Therefore, the altitude for the isosceles triangle is $\mathit{h}\mathbf{}\mathbf{=}\mathbf{}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{}\mathbf{-}\mathbf{}\frac{\mathbf{1}}{\mathbf{4}}{\mathbf{y}}^{\mathbf{2}}}$, where x represents the side lengths, y represents the base, and h represents the altitude.

Find the altitude of an isosceles triangle, if the base is $3inches$ and the length of two equal sides is $5inches$.

Solution: According to the formula of altitude for the isosceles triangle, we have $x=5,y=3$.

Altitude for an isosceles triangle: $h=\sqrt{{\mathrm{x}}^2-\frac{1}{4}{\mathrm{y}}^2}$

$$\begin{gathered} =\sqrt{{\left(5\right)}^2-\frac{1}{4}{\left(3\right)}^2} \\ =\frac{\sqrt{91}}{2} \end{gathered}$$

So, the altitude for the given isosceles triangle is $\frac{\sqrt{91}}{2}inches.$

Altitude Calculation for Different Triangles

A right triangle is a triangle with one angle as $90°$, and the altitude from one of the vertices to the hypotenuse can be explained with help from an important statement called the Right Triangle Altitude Theorem. This theorem gives the altitude formula for the right triangle.

Let's understand the theorem first.

Right Triangle Altitude Theorem: The height from the right angle vertex to the hypotenuse equals the square root of the product of the two segments of the hypotenuse.

Proof: From the given figure AC is the altitude of the right-angle triangle $△ABD$. Now using the Right Triangle Theorem, we get that two triangles $△ACD$ and $△ACB$ are similar.

Right Triangle Theorem: Drawing an altitude from the right angle vertex to the hypotenuse side of a right triangle results in two new triangles that are similar to the original triangle and to each other.

$∆ACD~∆ACB.$

$$\begin{gathered} \Rightarrow \frac{DC}{AC}=\frac{AC}{CB} \\ \Rightarrow A{C}^2=\hspace{0.17em}DC\times CB \\ \Rightarrow h^2=xy \\ \therefore h=\sqrt{xy} \end{gathered}$$

Hence from the above theorem, we can derive the formula for altitude.

Altitude for a right triangle$\mathit{h}\mathbf{}\mathbf{=}\sqrt{\mathbf{x}\mathbf{y}}$, where x and y are the lengths on either side of the altitude which together make up the hypotenuse.

In the given right triangle$∆ABC$, $AD=\hspace{0.17em}3cm$ and $DC=6cm.$ Find the length of altitude BD in the given triangle.

Solution: We will use the Right Angle Altitude Theorem to calculate the altitude.

Altitude for right triangle:$h=\sqrt{xy}$

$=\sqrt{3\times 6}=3\sqrt{2}$

Hence the length of the altitude for the right triangle is$3\sqrt{2}cm.$

Note: We cannot use the Pythagoras' theorem to calculate the altitude of the right triangle as not enough information is provided. So, we use the Right Triangle Altitude Theorem to find the altitude.

Altitude formula for an equilateral triangle

The equilateral triangle has all sides and angles equal. The altitude formula can be obtained using either Heron's formula or Pythagoras' formula. The altitude of an equilateral triangle is also known as a median.

Area of a triangle ∆ABC (by Heron's formula) = √(s(s-x)(s-y)(s-z))

And we also know that Area of triangle = 1/2 * b * h

So using both the above equations we get:

h = 2 * √(s(s - a)(s - b)(s - c)) / ba

Now the perimeter of an equilateral triangle is 3x. So semiperimeter s = 3x/2, and all the sides are equal.

h = 2√(3x^2) / x = 2x√3 / 2 = √3x

Altitude for equilateral triangle: h = √3x^2, where h is the altitude and x is the length for all three equal sides.

For an equilateral triangle $∆XYZ$, where XY, YZ, and ZX are equal sides with a length of $10cm.$, the altitude can be calculated as follows:

Solution: Given that $x=10cm,$ we can apply the formula for the altitude of an equilateral triangle:

Altitude for an equilateral triangle: $h=\hspace{0.17em}\frac{\sqrt{3}\mathrm{x}}{2}=\hspace{0.17em}\frac{\sqrt{3}\times 10}{2}=5\sqrt{3}$

Therefore, the length of the altitude for this equilateral triangle is $5\sqrt{3}cm.$

Concurrency of Altitudes

We know that all three altitudes of a triangle intersect at a point known as the orthocenter. Let's explore the concept of concurrency and the position of the orthocenter in different types of triangles.

In any triangle, the three altitudes are concurrent, meaning they intersect at a point called the orthocenter.

The coordinates of the orthocenter can be determined using the vertex coordinates of the triangle.

Position of the Orthocenter in a Triangle

The orthocenter's location changes depending on the triangle type and its altitudes.

Acute Triangle

The orthocenter of an acute triangle is located within the triangle.

Right Triangle

The orthocenter of a right triangle is located at the right angle vertex.

Right triangle Orthocenter

Obtuse Triangle

In an obtuse triangle, the orthocenter is located outside the triangle.

Obtuse triangle Orthocenter

Applications of Altitude

Altitude has various applications in a triangle:

1. Determining the orthocenter of a triangle.
2. Calculating the area of a triangle.

Altitude - Key takeaways

• A perpendicular segment from a vertex to the opposite side (or line containing the opposite side) is called an altitude of the triangle.
• A triangle possesses three altitudes, which can be situated outside, inside, or on the sides of the triangle.
• Altitude for scalene triangle is: $h=\frac{2\left(\sqrt{\mathrm{s}(\mathrm{s}-\mathrm{x})(\mathrm{s}-\mathrm{y})(\mathrm{s}-\mathrm{z})}\right)}{\mathrm{b}}$.
• Altitude for the isosceles triangle is: $h=\sqrt{{\mathrm{x}}^2-\frac{1}{4}{\mathrm{y}}^2}$.
• Altitude for a right triangle is: $h=\sqrt{\mathrm{xy}}$.
• Altitude for equilateral triangle is: $h=\hspace{0.17em}\frac{\sqrt{3}\mathrm{x}}{2}$.
• All the three altitudes of a triangle are concurrent; that is, they intersect at a point called the orthocenter.