Altitude of right triangle

In this chapter we will derive the formula to calculate the length of altitude of right triangle.

Formula for altitude length – Right triangle

The triangle in which one angle measure 90 degree is called right angle triangle.

Given below is the right triangle ABC with ∠B = 90 degree. The line BM is the altitude of the triangle which intersect hypotenuse AC at right angle.

Calculate altitude of right triangle

The altitude intersecting the hypotenuse divides the triangle into two similar triangle.

Hence, \mathtt{\triangle ABM\ \sim \ \triangle MBC}

In similar triangle, the corresponding sides are proportional.

\mathtt{MB^{2} =\ AB\ \times \ BC}\\\ \\ \mathtt{h^{2} =\ AB\ \times \ BC}\\\ \\ \mathtt{h\ =\ \sqrt{AB.\ BC}}

Hence, the altitude of right triangle is the square root of the product of two sides of the triangle.

I hope you understood the formula, let us solve some problems for further clarity.

Altitude of right triangle – Solved problems

Example 01
Given below is the right triangle ABC. Find the length of altitude BM.

Formula to calculate altitude of right triangle

In the above triangle;
AB = 6 cm
BC = 8 cm

The formula for length of altitude is given as;

\mathtt{h\ =\ \sqrt{AB.\ BC}}

Putting the values;

\mathtt{h\ =\ \sqrt{8\times 6}}\\\ \\ \mathtt{h\ =\ \sqrt{48}}\\\ \\ \mathtt{h\ =\ 4\ \sqrt{3} \ cm}

Hence, altitude length is \mathtt{\ 4\ \sqrt{3} \ cm\ } cm.

Example 02
Find the length of altitude BP of below right angled triangle.

Right triangle altitude

Given above is right angle triangle ABC where;
AB = 5 cm
AC = 13 cm

To find the length of altitude BP, we need to first find length of BC.

Applying Pythagoras theorem

\mathtt{AC^{2} =\ AB^{2} +\ BC^{2}}\\\ \\ \mathtt{13^{2} =\ 5^{2} +\ BC^{2}}\\\ \\ \mathtt{169\ =\ 25\ +\ BC^{2}}\\\ \\ \mathtt{BC^{2} =\ 169\ -\ 25}\\\ \\ \mathtt{BC^{2} =\ 144}\\\ \\ \mathtt{BC=\ 12}

Now using the altitude formula for right triangle.

\mathtt{h\ =\ \sqrt{AB.\ BC}}\\\ \\ \mathtt{h\ =\ \sqrt{5\times 12}}\\\ \\ \mathtt{h\ =\sqrt{60}}\\\ \\ \mathtt{h\ =\ 2\sqrt{15} \ cm}

Hence, length of altitude is \mathtt{2\sqrt{15} \ cm\ }

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