A Question on Rightangle Triangle and Find length of a Side

On a Right angled triangle, number of theorem are derived. One such theorem is Mid point Theorem, with below example theorem is very well explained.

Topic : Right angle Triangle and Mid Point Theorem

Problem : Given angle C is a right angle, E is a midpoint of AC, F is the midpoint of BC, AF = √41, BE = 2√26, Find AB

Solution :













From the figure,
In ∆ACF
AC² + CF² = AF² (by Pythagoras Theorem)
AC² + (CB/2)² = (√41)² (as CF is half of CB)
AC² + CB²/4 = 41

Now in ∆ECB
EC² + CB² = EB²
(AC/2)² + CB² = (2√26)² (as EC is half of AC)
AC²/4 + CB² = 104

Now adding both the equations, we get
AC² + CB²/4 + AC²/4 + CB² = 41 + 104
(1+1/4)AC² + (1+1/4)CB² = 145
5AC²/4 + 5CB²/4 = 145
5/4(AC² + CB²) = 145
Now, in ∆ACB, AC² + CB² = AB²

So we get, 5/4 AB² = 145
AB² = 145 * 4/5
AB² = 116
AB = √116
AB = 2√29

Hope the above elaborated explanation will help you to understand mid point theorem and help you to solve similar kind of problems.

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