Altitude of a Triangle in Geometry: Definitions, Methods, and Uses
Altitude of a Triangle in Geometry: In Geometry, the Altitude of a Triangle is the perpendicular line segment drawn from the vertex to the opposite side of the triangle. Based on the types of Triangles, it may lie inside or outside the triangle. The Altitude of a Triangle basically tells us about the height, when students have to measure the area of a triangle, concerning the base.
What is the Altitude of a Triangle?
The Altitude of a Triangle is the perpendicular distance from a vertex to the line containing the opposite side, often referred to as the base. It represents the height of the triangle relative to that particular base and is always drawn at a right angle (90°) to the base. The altitude can be located either inside or outside the triangle, depending on the type of triangle. In an acute triangle, all altitudes lie inside the triangle, whereas in a right triangle, two of the altitudes coincide with the legs of the triangle. In an obtuse triangle, one of the altitudes will fall outside the triangle. The altitude is essential for calculating the area of a triangle using the formula:
A = ½ × b × h
where b is the base and h is the height or altitude. This concept is useful not only in finding the area of triangles but also in solving other geometric problems related to their properties. Overall, altitude plays a crucial role in understanding the geometry of triangles and their spatial relationships.
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Altitude of a Triangle Formula
Students who want to find the Altitude of a Triangle can refer to the below formula
The area A of a triangle is given by the formula:
A = ½ × base × altitude
Rearranging this formula to solve for the altitude (h):
h = 2A/b
Where:
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A is the area of the triangle,
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b is the length of the base,
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h is the altitude (height) corresponding to that base.
How to Use the Altitude of a Triangle Formula:
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If the area A and the base b of the triangle are known, simply divide twice the area by the base length to find the altitude.
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If the area is not given, you can calculate it using Heron’s formula or other methods based on the triangle's sides and angles, and then use this formula to find the altitude.
Example:
For a triangle with an area of 30 square units and a base of 10 units:
h=60/10=6units
Thus, the altitude of the triangle is 6 units.
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Properties of Altitude of Triangle
Altitude of Triangles of various types has some properties. Students can check the properties below.
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The altitude is always perpendicular (forms a 90° angle) to the base of the triangle.
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It is drawn from a vertex to the line containing the opposite side (base), and the altitude can be inside or outside the triangle depending on its type.
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In an acute triangle, all altitudes lie inside the triangle.
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In a right triangle, two altitudes are the legs, and the third is drawn to the hypotenuse.
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In an obtuse triangle, one altitude falls outside the triangle.
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The altitude is used to calculate the area of the triangle with the formula
A = ½ × base × altitude
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The three altitudes of a triangle are concurrent and meet at a common point called the orthocenter. Its position varies depending on the type of triangle (inside for acute, at the right-angle vertex for right, and outside for obtuse triangles).
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Altitude of Scalene Triangle
In a scalene triangle, all sides and angles are different, and the altitudes are perpendicular from the vertices to the opposite sides and also have different lengths.
Consider a scalene triangle with sides a=6 cm, b=, and c=10 cm, and an area of 24 cm2. To find the altitude corresponding to the base b=8 cm:
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Use the area formula: A = ½ × base × altitude
24=1/2×8×h
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Solve for h: h = 24/4=6cm
So, the altitude corresponding to the base of 8 cm is 6cm.
This shows that in a scalene triangle, the altitude can be found using the area and the base, and each altitude corresponds to a different side.
Altitude of Obtuse Triangles
In an obtuse triangle, one of the altitudes falls outside the triangle, as the perpendicular from the vertex with the obtuse angle doesn’t intersect the triangle's interior.
Consider an obtuse triangle with side lengths a = 7 cm, b = 10 cm, and c = 12 cm, and an area of 36 cm2. To find the altitude corresponding to base b = 10 cm.
Using the area formula:
A = ½ × base × altitude
36 = ½ × 10 × h
h = 72/10 = 7.2 cm
Thus, the altitude is 7.2 cm, and it falls outside the triangle due to the obtuse angle.
Altitude of Equilateral Triangle
In an equilateral triangle, all sides and angles are equal. The altitude is the perpendicular from a vertex to the opposite side and also bisects the base.
Formula for Altitude:
h=√3/2 × s
Where s is the side length.
Example:
For an equilateral triangle with side length s = 6 cm
h=√3/2 × 6 = 3√3 = 5.2 cm
Thus, the altitude is approximately 5.2 cm
Altitude of Right Triangle
In a right triangle, one of the angles is 90∘90^\circ90∘, and the altitude can be easily understood because two of the altitudes coincide with the legs of the triangle.
In the above figure,
△ADB ∼ △BDC
Thus,
AD/BD = BD/DC
BD2 = AD.DC
h2 = x.y
h = √xy
Hence, is the altitude of a right triangle.
Altitude of an Isosceles Triangle
The altitude of an isosceles triangle is the perpendicular segment from the vertex opposite the base to the midpoint of the base. It bisects the base and divides the triangle into two equal right triangles.
Formula:
h= √a2 - (b/2)2
Where a is the equal side and b is the base
Example:
For an isosceles triangle with equal sides a = 5 cm and base b = 6 cm
h= √52- (6/2)2 = √25-9 = √16 = 4 cm
Thus, the altitude is 4 cm.
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Difference Between Median and Altitude
Students can check the difference between Median and Altitude in the table below.
|
Feature |
Median |
Altitude |
|
Definition |
A line segment connecting a vertex to the midpoint of the opposite side. |
A perpendicular line segment from a vertex to the opposite side (or its extension). |
|
Purpose |
Divides the triangle into two smaller triangles of equal area. |
Represents the height of the triangle; used for area calculation. |
|
Perpendicularity |
Not necessarily perpendicular to the opposite side. |
Always perpendicular to the opposite side. |
|
Intersection Point |
The three medians intersect at the centroid (divides each median in a 2:1 ratio). |
The three altitudes intersect at the orthocenter. |
|
Role in Area |
Does not directly help in area calculation. |
Essential for area calculation: Area = ½ × base × altitude |
|
Symmetry |
In an isosceles triangle, the median bisects the base. |
In an isosceles triangle, the altitude also bisects the base. |
|
Example |
In a scalene triangle, the median and altitude are usually different. |
In a scalene triangle, the altitude and median are typically different. |
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Altitude of Triangles FAQs
Q1. What is the altitude of a triangle?
Ans. The altitude of a triangle is the perpendicular distance from a vertex to the line containing the opposite side (also called the base).
Q2. How do you find the altitude of a triangle?
Ans. To find the altitude, use the formula Area = ½ × base × altitude, and solve for the altitude.
Q3. What is the difference between altitude and height in a triangle?
Ans. Height and altitude refer to the same concept: the perpendicular distance from a vertex to the opposite side.
Q4. What is the altitude of a right triangle?
Ans. In a right triangle, the altitudes corresponding to the two legs are the lengths of the legs, and the altitude to the hypotenuse can be calculated using the area formula.
Q5. What is the altitude of an isosceles triangle?
Ans. In an isosceles triangle, the altitude is drawn from the vertex opposite the base to the midpoint of the base, bisecting it.
