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

Prove that in a right triangle,the square of hypotenuse is equal to the sum of squares of the other two sides.

Accepted Answer

Let $△ A B C$ be a right triangle and $A D$ is a perpendicular from $A$ on $B C$ .

Pythagoras theorem states that, in a right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.

To prove: $B C^{2} = A B^{2} + A C^{2}$ .

Proof: In $△ A B C$ and $△ A C D$ ,

$∠ A C B = ∠ A C D$ [common angle]
$∠ C A B = ∠ A D C$ [each $90 °$ ]
So by $A A$ similarity $△ A B C ~ △ A C D$ ... $(1)$
So, $\frac{A C}{C D} = \frac{A B}{A D} = \frac{B C}{A C}$ .

Now, on taking first and third results, we get
$\frac{A C}{C D} = \frac{B C}{A C}$
$∴ A C^{2} = B C \times C D$ ... $(2)$

Now, in $△ A B C$ and $△ A B D$ ,
$∠ A B C = ∠ A B D$ [common angle]
$∠ C A B = ∠ A D B$ [each $90 °$ ]
So, by $A A$ similarity $△ A B C ~ △ A B D$ ... $(3)$
Hence, $\frac{A C}{A D} = \frac{A B}{B D} = \frac{B C}{A B}$ .

On taking second and third result, we get
$\frac{A B}{B D} = \frac{B C}{A B}$ .
On cross multiplying, $A B^{2} = B C \times B D$ ... $(4)$

On adding equation number $(2) a n d (4)$ ,
$A C^{2} + A B^{2} = B C \times C D + B C \times B D$
$\Rightarrow A C^{2} + A B^{2} = B C \times [C D + B D]$
$A C^{2} + A B^{2} = B C \times B C [∵ C D + B D = B C]$
$A C^{2} + A B^{2} = B C^{2}$ ... $(5)$

Hence, proved.

Prove that in a right triangle,the square of hypotenuse is equal to the sum of squares of the other two sides.

Important Questions on Triangles

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