Altitude

What is altitude?

The altitude is measured as the distance from the vertex to the base and so it is also known as the height of a triangle. Every triangle has three altitudes, and these altitudes may lie outside, inside, or on the side of a triangle. Let's take a look at how it may look.

Altitudes with different positions, ck12.org

Properties of an altitude

Here are some of the properties of altitude:

• An altitude makes an angle of $90°$on the side opposite from the vertex.
• The location of altitude changes depending on the type of triangle.
• As the triangle has three vertices, it has three altitudes.
• The point where these three altitudes intersect is called the orthocenter of the triangle.

Altitude formula for different triangles

There are different forms of altitude formulas based on the type of triangle. We will look at the altitude formula for triangles in general as well as specifically for scalene triangles, isosceles triangles, right triangles, and equilateral triangles, including brief discussions of how these formulas are derived.

General altitude formula

As altitude is used to find 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 is the base of triangle and h is the height/ altitude. So from this, we can deduce 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)$\mathbf{=}\mathbf{(}\mathbf{2}\mathbf{\times }\mathbf{A}\mathbf{r}\mathbf{e}\mathbf{a}\mathbf{)}\mathbf{/}\mathit{b}$

Altitude formula for scalene triangle

The triangle which has different side lengths for all three sides is known as the scalene triangle. Here Heron's formula is used to derive the altitude.

Altitude for scalene triangle, StudySmarter Originals

Area of a triangle$∆ABC$(by Heron's formula)$=\sqrt{s\left(s-x\right)\left(s-y\right)\left(s-z\right)}$

Here s is the semi perimeter of the triangle (i.e., $s=\frac{\left(x+y+z\right)}{2}$) and x, y, z are the lengths of sides.

Now using the general formula of the area and equating it with Heron's formula we can obtain the altitude,

Area$=\frac{1}{2}\times b\times h$

$\Rightarrow \sqrt{s\left(s-x\right)\left(s-y\left(s-z\right)\right)}=\frac{1}{2}\times b\times h$

So, the altitude for a scalene triangle: $\mathit{h}\mathbf{=}\frac{\mathbf{2}\mathbf{\left(}\sqrt{\mathbf{s}\mathbf{(}\mathbf{s}\mathbf{-}\mathbf{x}\mathbf{)}\mathbf{(}\mathbf{s}\mathbf{-}\mathbf{y}\mathbf{)}\mathbf{(}\mathbf{s}\mathbf{-}\mathbf{z}\mathbf{)}}\mathbf{\right)}}{\mathbf{b}}\mathbf{.}$

Altitude formula for isosceles triangle

An isosceles triangle is a triangle whose two sides are equal. The altitude of an isosceles triangle is the perpendicular bisector of that triangle with its opposite side. We can derive its formula using the properties of the isosceles triangle and Pythagoras' theorem.

Altitude in Isosceles triangle, StudySmarter Originals

As triangle$∆ABC$ is an isosceles triangle, sides $AB=AC$with length x. Here we use one of the properties for an isosceles triangle, which states that the altitude bisects its base side into two equal parts.

$\Rightarrow \frac{1}{2}BC=DC=BD$

Now applying Pythagoras' theorem on$∆ABD$ we get:

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

Now substituting all the values of the given side we get:

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

Hence, 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 is the side lengths, y is the base, and h is the altitude.

Altitude formula for right triangle

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.

Right triangle altitude, StudySmarter Originals

Let's understand the theorem first.

Right Triangle Altitude Theorem: The altitude from the right angle vertex to the hypotenuse is equal to the geometric mean 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 Similarity Theorem, we get that two triangles $△ACD$ and $△ACB$ are similar.

$∆ACD~∆ACB.$

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

Hence from the above theorem, we can get 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.

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 equilateral triangle

The equilateral triangle is a triangle with all sides and angles equal respectively. We can derive the formula of altitude by using either Heron's formula or Pythagoras' formula. The altitude of an equilateral triangle is also considered a median.

Equilateral triangle altitude, StudySmarter Originals

Area of a triangle$∆ABC$(by Heron's formula)$=\sqrt{s\left(s-x\right)\left(s-y\right)\left(s-z\right)}$

And we also know that Area of triangle $=\frac{1}{2}\times b\times h$

So using both the above equation we get:

$h=\frac{2\left(\sqrt{s(s-a)(s-b)(s-c)}\right)}{base}$

Now the perimeter of an equilateral triangle is 3x. So semiperimeter $s=\frac{3x}{2}$, and all the sides are equal.

$$\begin{gathered} h=\frac{2\sqrt{\frac{3x}{2}\left(\frac{3x}{2}-x\right)\left(\frac{3x}{2}-x\right)\left(\frac{3x}{2}-x\right)}}{x} \\ =\frac{2\sqrt{\left(\frac{3x}{2}\right)\left(\frac{x}{2}\right)\left(\frac{x}{2}\right)\left(\frac{x}{2}\right)}}{x} \\ =\frac{2}{x}\times x^2\frac{\sqrt{3}}{4} \\ =\frac{\sqrt{3}x}{2} \end{gathered}$$

Altitude for equilateral triangle:$\mathit{h}\mathbf{}\mathbf{=}\mathbf{\hspace{0.17em}}\frac{\sqrt{\mathbf{3}}\mathbf{x}}{\mathbf{2}}$, where h is the altitude and x is the length for all three equal sides.

Concurrency of altitudes

We discussed in the properties of altitude that all three altitudes of a triangle intersect at a point called the orthocenter. Let's understand the concepts of concurrency and orthocenter position in different triangles.

We can calculate the coordinates of the orthocenter using the vertex coordinates of the triangle.

Position of the orthocenter in a triangle

The position of the orthocenter may vary depending on the type of triangle and altitudes.

Acute Triangle

The orthocenter in an acute triangle lies inside the triangle.

Acute triangle Orthocenter, StudySmarter Originals

Right Triangle

The orthocenter of the right triangle lies on the right angle vertex.

Right triangle Orthocenter, StudySmarter Originals

Obtuse Triangle

In an obtuse triangle, the orthocenter lies outside the triangle.

Obtuse triangle Orthocenter, StudySmarter Originals

Applications of Altitude

Here are a few applications of altitude in a triangle:

1. The foremost application of altitude is to determine the orthocenter of that triangle.
2. Altitude can also be used to calculate the area of a triangle.