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## What is altitude?

A perpendicular segment from a vertex to the opposite side – or line containing the opposite side – is called an **altitude** of the triangle.

Triangles with altitude, StudySmarter Originals

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.

## Properties of an altitude

Here are some of the properties of altitude:

- An altitude makes an angle of $90\xb0$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:

$Area=\frac{1}{2}\times b\times h\phantom{\rule{0ex}{0ex}}\Rightarrow 2\times Area=b\times h\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{2\times Area}{b}=h$

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

For a triangle$\u2206ABC$, the area is$81c{m}^{2}$with a base length of$9cm$. Find the altitude length for this triangle.

__Solution__: Here we are given the area and base for the triangle$\u2206ABC$. So we can directly apply the general formula to find the length of altitude.

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

### 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.

**Heron's formula** is the formula to find the area of a triangle based on the length of sides, perimeter, and semi-perimeter.

Area of a triangle$\u2206ABC$(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$=\frac{1}{2}\times b\times h$

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

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

In a scalene triangle$\u2206ABC$, AD is the altitude with base BC. The length of all three sides AB, BC, and AC are 12, 16, and 20, respectively. The perimeter for this triangle is given as 48 cm. Calculate the length of the altitude AD.

**Solution****:** Here$x=12cm,y=16cm,z=20cm$are given. Base BC has a length of 16 cm. To calculate the length of altitude, we need a semiperimeter. Let's first find the value of the semiperimeter from the perimeter.

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

Now we can apply the formula of altitude to get the measure of altitude.

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

$=\frac{2\sqrt{24(24-12)(24-16)(24-20)}}{16}\phantom{\rule{0ex}{0ex}}=\frac{2\times 96}{16}=12$

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

### 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.

As triangle$\u2206ABC$ 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$\u2206ABD$ we get:

$A{B}^{2}=A{D}^{2}+B{D}^{2}\phantom{\rule{0ex}{0ex}}\Rightarrow A{B}^{2}=A{D}^{2}+{\left(\frac{1}{2}BC\right)}^{2}\phantom{\rule{0ex}{0ex}}\Rightarrow A{D}^{2}=\hspace{0.17em}A{B}^{2}-{\left(\frac{1}{2}BC\right)}^{2}$

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

$\Rightarrow {h}^{2}={x}^{2}-\frac{1}{4}{y}^{2}\phantom{\rule{0ex}{0ex}}\therefore h=\sqrt{{\mathrm{x}}^{2}-\frac{1}{4}{\mathrm{y}}^{2}}$

Hence, the **a****ltitude 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.

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:$h=\sqrt{{\mathrm{x}}^{2}-\frac{1}{4}{\mathrm{y}}^{2}}$

$=\sqrt{{\left(5\right)}^{2}-\frac{1}{4}{\left(3\right)}^{2}}\phantom{\rule{0ex}{0ex}}=\frac{\sqrt{91}}{2}$

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

### Altitude formula for right triangle

A right triangle is a triangle with one angle as$90\xb0$, 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 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 $\u25b3ABD$. Now using the Right Triangle Similarity Theorem, we get that two triangles $\u25b3ACD$ and $\u25b3ACB$ are similar.

**Right Triangle Similarity Theorem:** If an altitude is drawn from the right angle vertex to the hypotenuse side of the right triangle, then the two new triangles formed are similar to the original triangle and are also similar to each other.

$\u2206ACD~\u2206ACB.$

$\Rightarrow \frac{DC}{AC}=\frac{AC}{CB}\phantom{\rule{0ex}{0ex}}\Rightarrow A{C}^{2}=\hspace{0.17em}DC\times CB\phantom{\rule{0ex}{0ex}}\Rightarrow {h}^{2}=xy\phantom{\rule{0ex}{0ex}}\therefore h=\sqrt{xy}$

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.**

In the given right triangle$\u2206ABC$, $AD=\hspace{0.17em}3cm$ 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 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.

Area of a triangle$\u2206ABC$(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.

$h=\frac{2\sqrt{\frac{3x}{2}\left(\frac{3x}{2}-x\right)\left(\frac{3x}{2}-x\right)\left(\frac{3x}{2}-x\right)}}{x}\phantom{\rule{0ex}{0ex}}=\frac{2\sqrt{\left(\frac{3x}{2}\right)\left(\frac{x}{2}\right)\left(\frac{x}{2}\right)\left(\frac{x}{2}\right)}}{x}\phantom{\rule{0ex}{0ex}}=\frac{2}{x}\times {x}^{2}\frac{\sqrt{3}}{4}\phantom{\rule{0ex}{0ex}}=\frac{\sqrt{3}x}{2}$

**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.

For an equilateral triangle$\u2206XYZ$, XY, YZ, and ZX are equal sides with the length of$10cm.$Calculate the length of the altitude for this triangle.

__Solution__: Here$x=10cm.$ Now we will apply the formula of altitude for an equilateral triangle.

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

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

## 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.

All three altitudes of a triangle are concurrent; that is, they intersect at a point. This point of concurrency is called the **orthocenter** of a triangle.

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.

#### Right Triangle

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

#### Obtuse Triangle

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

## Applications of Altitude

Here are a few applications of altitude in a triangle:

- The foremost application of altitude is to determine the orthocenter of that triangle.
- Altitude can also be used to calculate 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.
- Every triangle has three altitudes and these altitudes may lie outside, inside, or on the side of a 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.

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##### Frequently Asked Questions about Altitude

What is the altitude of a triangle?

A perpendicular segment from a vertex to the opposite side or line containing the opposite side is called an altitude of the triangle.

How to find the altitude of a triangle?

We can find the altitude of a triangle from the area of that triangle

What is the difference between median and altitude of a triangle?

Altitude is the perpendicular line segment from a vertex to opposite side. Whereas, median is a line segment from one vertex to the middle of opposite side.

What is the formula for finding the altitude of a triangle?

The general formula for altitude is as follows:

**Altitude (h)**.

What is the rules in finding the altitude of a triangle?

The rule of finding the altitude is to first identify the type of triangle.

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