How To Find The Altitude Of A Hypotenuse

Distance of a Triangle

The altitude of a triangle is a perpendicular that is drawn from the vertex of a triangle to the opposite side. Since at that place are 3 sides in a triangle, three altitudes tin be drawn in it. Dissimilar triangles have dissimilar kinds of altitudes. The altitude of a triangle which is besides called its height is used in calculating the area of a triangle and is denoted by the letter of the alphabet 'h'.

1.	Altitude of a Triangle Definition
2.	Distance of Triangle Backdrop
3.	Altitude of Triangle Formula
4.	Difference Between Median and Altitude of Triangle
5.	FAQs on Altitude of a 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. The altitude makes a right angle with the base of operations of the triangle that it touches. It is normally referred to every bit the peak of a triangle and is denoted by the alphabetic character 'h'. It can exist measured by calculating the distance between the vertex and its opposite side. It is to be noted that iii altitudes can exist fatigued in every triangle from each of the vertices. Notice the following triangle and encounter the signal where all the 3 altitudes of the triangle meet. This point is known every bit the 'Orthocenter'.

Altitude of Triangle

Altitude of a Triangle Properties

The altitudes of diverse types of triangles have some backdrop that are specific to certain triangles. They are equally follows:

A triangle tin 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° to the side reverse to it.
The indicate of intersection of the three altitudes of a triangle is called the orthocenter of the triangle.

Distance of a Triangle Formula

The bones formula to find the surface area of a triangle is: Expanse = 1/2 × base × acme, where the pinnacle represents the altitude. Using this formula, nosotros tin can derive the formula to calculate the height (distance) of a triangle: Altitude = (2 × Area)/base. Let the states acquire how to find out the altitude of a scalene triangle, equilateral triangle, right triangle, and isosceles triangle.

The important formulas for the distance of a triangle are summed upwards in the following table. The following section explains these formulas in detail.

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

Distance of a Scalene Triangle

A scalene triangle is ane in which all three sides are of unlike lengths. To find the altitude of a scalene triangle, we use the Heron's formula as shown here. \(h=\dfrac{two\sqrt{due south(south-a)(s-b)(south-c)}}{b}\) Here, h = elevation or altitude of the triangle, 's' is the semi-perimeter; 'a, 'b', and 'c' are the sides of the triangle.

Altitude of a Scalene Triangle

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

The area of a triangle using the Heron'due south formula is, \(Area= \sqrt{southward(due south-a)(due south-b)(s-c)}\).
The basic formula to detect the expanse of a triangle with respect to its base of operations 'b' and altitude 'h' is: Expanse = 1/2 × b × h
If we identify both the area formulas equally, we get, \[\begin{align} \dfrac{i}{two}\times b\times h = \sqrt{s(due south-a)(s-b)(s-c)} \end{align}\]
Therefore, the altitude of a scalene triangle is \[\begin{marshal} h = \dfrac{2\sqrt{due south(s-a)(s-b)(s-c)}}{b} \end{align}\]

Altitude of an Isosceles Triangle

A triangle in which ii sides are equal is called an isosceles triangle. The altitude of an isosceles triangle is perpendicular to its base.

Let us see the derivation of the formula for the altitude of an isosceles triangle. In the isosceles triangle given to a higher place, side AB = Ac, BC is the base, and Advertising is the altitude. Let us represent AB and AC equally 'a', BC every bit 'b', and AD as 'h'. One of the backdrop 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 △ADB, we get,

AD^two= AB²- BD² ....(Equation one)

Since, Advert is the bisector of side BC, information technology divides it into 2 equal parts.

So, BD = 1/2 × BC

Substitute the value of BD in Equation ane,

AD² = AB²- BD²

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

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

Altitude of an Equilateral Triangle

A triangle in which all three sides are equal is chosen an equilateral triangle. Because the sides of the equilateral triangle to exist 'a', its perimeter = 3a. Therefore, its semi-perimeter (southward) = 3a/2 and the base of the triangle (b) = a.

altitude of an equilateral triangle

Let us see the derivation of the formula for the distance 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, and so it volition exist written as 'a' in this case; s = semi perimeter of the triangle, which will be written equally 3a/2 in this example.

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

\(\begin{marshal} h=\dfrac{2}{a} \sqrt{\dfrac{3a}{2}(\dfrac{3a}{2}-a)(\dfrac{3a}{two}-a)(\dfrac{3a}{2}-a)} \stop{align}\)

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

\(\brainstorm{align} h=\dfrac{2}{a} \times \dfrac{a^ii\sqrt{3}}{iv} \terminate{marshal}\)

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

Altitude of a Correct Triangle

A triangle in which one of the angles is 90° is called a right triangle or a right-angled 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. Information technology is popularly known as the Right triangle altitude theorem.

Altitude of a Right Triangle

Allow us encounter the derivation of the formula for the distance of a right triangle. In the in a higher place figure, △PSR ∼ △RSQ

And then, \(\dfrac{PS}{RS}=\dfrac{RS}{SQ}\)

RSⁱⁱ= PS × SQ

Referring to the figure given above, this tin can also be written as: h² = x × y, here, 'x' and 'y' are the bases of the two like triangles: △PSR and △RSQ.

Therefore, the distance of a correct triangle (h) = √xy

Altitude of an Obtuse Triangle

A triangle in which one of the interior angles is greater than 90° is called an obtuse triangle. The altitude of an obtuse triangle lies outside the triangle. Information technology is usually fatigued by extending the base of the obtuse triangle every bit shown in the figure given beneath.

Altitude of an Obtuse Triangle

Difference Between Median and Altitude of Triangle

We know that the median and the altitude of a triangle are line segments that bring together the vertex to the contrary side of a triangle. All the same, they are different from each other in many ways. Notice the figure and the table given beneath to understand the difference between the median and altitude of a triangle.

Difference between median and altitude of triangle

Median of a Triangle	Altitude of a Triangle
The median of a triangle is the line segment drawn from the vertex to the opposite side.	The altitude of a triangle is the perpendicular distance from the base to the opposite vertex.
It ever lies inside the triangle.	Information technology can be both exterior or within the triangle depending on the blazon of triangle.
Information technology divides a triangle into two equal parts.	It does non dissever the triangle into ii equal parts.
It bisects the base of operations of the triangle into ii equal parts.	It does non bifurcate the base of the triangle.
The point where the iii medians of a triangle meet is known as the centroid of the triangle.	The point where the 3 altitudes of the triangle meet is known as the orthocenter of that triangle.

Important Notes

Here is a list of a few important points related to the altitude of a triangle.

The point where all the three altitudes of a triangle intersect is called the orthocenter.
Both the altitude and the orthocenter can lie inside or outside the triangle.
In an equilateral triangle, the distance is the same as the median of the triangle.

Topics Related to Distance of a Triangle

Bank check out some interesting topics related to the distance of a triangle.

Area
Basic Area Concepts
Triangles
Types of Triangles
Altitude of a Triangle Formula
Triangle Top Calculator

Altitude of a Triangle Examples

Example 1: The surface area of a triangle is 72 square units. Detect the length of the altitude if the length of the base is 9 units.

Solution:

We know that altitude of a triangle, h = (2 × Area) / Base.
Given, area = 72 square units and base of operations = 9 units.
Altitude 'h' = (ii × 72) / 9
= 144/9
= 16 units.
Therefore, distance 'h' = 16 units.
Example 2: Calculate the length of the altitude of a scalene triangle whose sides are 7 units, 8 units, and ix units respectively.

Solution:

The perimeter of a triangle is the sum of all the sides = 7 + 8 + nine = 24 units. Semi-perimeter (s) = 24/ii =12 units. Let united states name the sides of the scalene triangle to be 'a', 'b', and 'c' respectively. Therefore, a = 9 units, b = 8 units and c = 7 units;

The altitude of the triangle:

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

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

Altitude(h) = \(\frac{2 \sqrt{12\ \times three\ \times four\ \times v}}{8}\)

Altitude (h) = half-dozen.70 units
Example 3: Summate the altitude of an isosceles triangle whose two equal sides are 8 units and the third side is 6 units.

Solution:

The equal sides (a) = 8 units, the tertiary side (b) = half-dozen units. In an isosceles triangle the altitude is:

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

Altitude(h)= \(\sqrt{8^2-\frac{6^2}{4}}\)

Altitude(h)= √[64-(36/4)]

Distance(h)= √55

Distance(h)= 7.41 units

Therefore, the altitude of the isosceles triangle is seven.41 units.

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Practise Questions on Distance of a Triangle

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FAQs on Altitude of a Triangle

What is the Altitude of a Triangle in Math?

The altitude of a triangle is a line segment that is fatigued from the vertex of a triangle to the side opposite to it. It is perpendicular to the base or the opposite side which information technology touches. Since in that location are 3 sides in a triangle, 3 altitudes can be fatigued in a triangle. All the three altitudes of a triangle intersect at a bespeak chosen the 'Orthocenter'.

What is the Altitude of Triangle Formula?

The altitude of a triangle tin can be calculated according to the unlike formulas defined for the various types of triangles. The formulas used for the dissimilar triangles are given below:

Distance of a scalene triangle = \(h= \frac{2 \sqrt{southward(due south-a)(s-b)(southward-c)}}{b}\); where 'a', 'b', 'c' are the 3 sides of the triangle; 's' is the semi perimeter of the triangle.
Distance of an isosceles triangle = \(h= \sqrt{a^2- \frac{b^2}{4}}\); where 'a' is 1 of the equal sides, 'b' is the third side of the triangle.
Distance of an equilateral triangle = \(h= \frac{a\sqrt{3}}{ii}\); where 'a' is i side of the triangle
Distance of a right triangle = \(h= \sqrt{xy}\); where 'x' and 'y' are the bases of the two similar triangles formed.

What are the Properties of Altitude of a Triangle?

The distance of a triangle is the line fatigued from a vertex to the opposite side of a triangle. The of import properties of the altitude of a triangle are as follows:

A triangle can take 3 altitudes.
The altitudes can be within or outside the triangle, depending on the type of triangle.
The altitude makes an angle of ninety° to the side opposite to it.
The point of intersection of the three altitudes of a triangle is called the orthocenter of the triangle.

How to Discover the Altitude of a Right Triangle?

A triangle in which i of the angles is xc° is a right triangle. When an altitude is drawn from a vertex to the hypotenuse of a correct-angled triangle, information technology forms 2 similar triangles. The formula to calculate the altitude of a correct triangle is h =√xy. where 'h' is the altitude of the right triangle and 'x' and 'y' are the bases of the 2 like triangles formed after drawing the altitude from a vertex to the hypotenuse of the right triangle.

How to Find the Altitude of a Scalene Triangle?

A triangle in which all iii sides are diff is a scalene triangle. The formula to calculate the distance of a scalene triangle is \(h= \frac{2 \sqrt{s(due south-a)(s-b)(s-c)}}{b}\), where 'h' is the altitude of the scalene triangle; 's' is the semi-perimeter, which is half of the value of the perimeter, and 'a', 'b' and 'c' are three sides of the scalene triangle.

What is the Difference Betwixt Median and Distance of Triangle?

The distance of a triangle and median are two unlike line segments drawn in a triangle. The distance of a triangle is the perpendicular distance from the base to the opposite vertex. Information technology can be located either outside or inside the triangle depending on the blazon of triangle. The median of a triangle is the line segment fatigued from the vertex to the opposite side that divides a triangle into two equal parts. Information technology bisects the base of the triangle and always lies inside the triangle.

Does the Altitude of a Triangle Always Make 90° With the Base of the Triangle?

Yes, the altitude of a triangle is a perpendicular line segment drawn from a vertex of a triangle to the base or the side reverse to the vertex. Since it is perpendicular to the base of the triangle, it always makes a ninety° with the base of the triangle.

Is the Altitude of a Triangle Same as the Summit of a Triangle?

Yes, the altitude of a triangle is also referred to equally the height of the triangle. Information technology is denoted past the modest letter 'h' and is used to calculate the surface area of a triangle. The formula for the area of a triangle is (one/2) × base of operations × acme. Here, the 'elevation' is the altitude of the triangle.

Does the Distance of an Obtuse Triangle lie Inside the Triangle?

No, the distance of an obtuse triangle lies outside the triangle because the angle opposite to the vertex from which the distance is fatigued is an obtuse angle. This is done past extending the base of the given obtuse triangle.