In this chapter, we will learn about the concept of altitude of triangle in detail with examples.
The chapter also contain some formulas for calculating lengths of altitude in different types of triangles.
At the end some solved problems are provided for practice purpose.
What is altitude of triangle ?
The line segment that joins the vertex and opposite side at 90 degree angle is called altitude of triangle.
Consider the below triangle ABC.
Note that line AM joins the vertex A and side BC at 90 degree angle. Hence, the segment AM is the altitude of triangle.
Since the altitude intersect the other side at 90 degree angle, we can say that;
\mathtt{\angle AMC=\ \angle AMB=\ 90\ degree}
There can be only three altitude in any given triangle which may be present inside and outside the triangle.
Given below is the image of triangle with all the altitudes.
In the above triangle ABC; line segments AM, BN and CQ are the altitudes.
Property of altitude of triangle
Given below are important properties of triangle’s altitude;
(a) Altitude of triangle joins the vertex and opposite side of triangle at right triangle.
(b) There can be only three altitudes in the given triangle.
(c) The point at which all the altitude intersects is called Orthocenter of triangle.
Given above is the triangle ABC with AP, BQ & AR as the altitude of triangle.
Observe that all the altitude intersect at point O, hence is the orthocenter of triangle.
(d) Orthocenter can lie inside and outside the triangle.
In acute triangle, orthocenter lie inside the triangle.
In obtuse triangle, orthocenter lie outside the triangle.
In right triangle, orthocenter lie at the vertex.
(e) Altitude helps in finding area of triangle.
If you know the length of altitude and the base, then area of triangle can be easily calculated using the following formula;
\mathtt{Area\ =\frac{1}{2} \times \ base\ \times height}
(f) Formula for length of altitude
Given below are the shortcut formula for length of triangle in different types of triangle.
Altitude formula for scalene triangle
In scalene triangle, all the sides have different base.
Given below is the triangle ABC, with sides “a”, “b” and “c”.
The formula for altitude is given as;
\mathtt{h=\frac{2\sqrt{s( s-a)( s-b)( s-c)}}{base\ length}}
For the above triangle, base length is “c”
Altitude formula for isosceles triangle
The triangle in which two sides are equal are called isosceles triangle.
Given below is the isosceles triangle ABC with equal sides AB = AC and altitude AM.
The formula for altitude is given as;
\mathtt{h=\sqrt{a^{2} -\frac{1}{4} b^{2}}}
Altitude of equilateral triangle
The triangle in which all sides are equal are called equilateral triangle.
Given below is the equilateral triangle where AB = BC = CA.
The formula for altitude of triangle is given as;
\mathtt{h=\frac{\sqrt{3}}{2} a}
(g) Altitude of Obtuse triangle
The triangle in which one of the interior angle is greater than 90 degree is called obtuse triangle.
In obtuse triangle, the altitude extends outside the triangle.
The above triangle ABC is an obtuse triangle.
Note that the altitude AO is outside the triangle.
Length of altitude of obtuse triangle is given as;
\mathtt{H\ =\ \frac{2}{Base}\sqrt{s( s-a)( s-b)( s-c)}}
Where;
S = (a + b + c) / 2
I hope you understood the concepts and properties. Given below are solved examples for your practice.
Altitude of Triangle – Solved Problems
Example 01
Given below is the isosceles triangle ABC with sides AB = AC = 5 cm and side BC = 9 cm. Find the length of altitude AM.
Solution
As discussed above, the formula for height of isosceles triangle is;
\mathtt{h=\sqrt{a^{2} -\frac{1}{4} b^{2}}}
Putting the values;
\mathtt{h\ =\ \sqrt{5^{2} -\frac{9^{2}}{4}}}\\\ \\ \mathtt{h=\ \sqrt{25-\frac{81}{4}}}\\\ \\ \mathtt{h=\ \sqrt{\frac{100-81}{4}}}\\\ \\ \mathtt{h\ =\ \sqrt{\frac{19}{4}}}\\\ \\ \mathtt{h\ =\frac{\sqrt{19}}{2}}
Hence, \mathtt{h\ =\frac{\sqrt{19}}{2}} is the solution.
Example 02
Given below is the equilateral triangle with side length 8 cm. Find the length of altitude.
Solution
As discussed above, the formula for altitude of equilateral triangle is given as;
\mathtt{h=\frac{\sqrt{3}}{2} a}
Putting the values;
\mathtt{h\ =\ \frac{\sqrt{3}}{2} \times 8}\\\ \\ \mathtt{h=4\sqrt{3}}
Hence, the length of altitude is \mathtt{h=4\sqrt{3}} cm.
Example 03
The length of altitude in isosceles triangle is 8 cm and length of base is 12 cm. Find the length of other two sides.
Solution
In the above triangle ABC.
AM = 8 cm
BC = 12 cm
In isosceles triangle, the altitude divides the base into two equal parts.
Hence, BM = MC = 6 cm.
Applying Pythagoras theorem in triangle ABM.
\mathtt{AB^{2} =\ BM^{2} +AM^{2}}\\\ \\ \mathtt{AB^{2} =\ 6^{2} +8^{2}}\\\ \\ \mathtt{AB^{2} =\ 36\ +\ 64}\\\ \\ \mathtt{AB^{2} =\ 100}\\\ \\ \mathtt{AB\ =\ 10}
Hence, length of side AB & AC = 10 cm