Name: Example on Thales Theorem
Uploaded: 2019-07-19
Duration: 4 min 26 s
Description: If the midpoint of two sides of a triangle is joined, then this line segment is parallel to the third side. Let's explore more by an example in this video.

Question

Draw a right triangle and the perpendicular from the midpoint of the hypotenuse to the base. Prove that the circumcentre of a right triangle is the midpoint of its hypotenuse.

Accepted Answer

Given, $M$ is the midpoint of $A C$ .

In figure $M N, C B$ are perpendiculars to $A B$

And $M N, C B$ are parallel lines.

Here, $M N$ divides $A B and A C$ in the same ratio.

Here, $M$ is the midpoint of the $A C$ . So, $N$ is the midpoint of $A B$ .

$∴ A N = N B$

Now, in figure $∠ A N M = ∠ B N M = 90 °$

$M N = M N$ (Common side)

$Δ A N M ≅ Δ B N M$ (By Side-Angle-Side congruence condition)

$A M = M B$ ……. (1)

$M$ is the midpoint of $A C$

$A M = M C$ ……. (2)

From (1) and (2), $A M = M B = M C$

So, in the large triangle, the distances from the midpoint of the hypotenuse to all the vertices are equal.

i.e. the points $A, B and C$ are at same distance from $M$ . So, a circle can be drawn with $M$ as centre and pass through these three points. So this circle is the circumcircle of $Δ A B C$ .

Hence, the circumcentre of a right triangle is the midpoint of its hypotenuse.

Draw a right triangle and the perpendicular from the midpoint of the hypotenuse to the base. Prove that the circumcentre of a right triangle is the midpoint of its hypotenuse.

Important Questions on Parallel Lines

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