LEARNATHON
III

Competition for grade 6 to 10 students! Learn, solve tests and earn prizes!

Theory:

Statement:
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Explanation:

The theorem states that in the right angled triangle $$ABC$$, $$AC^2=AB^2+BC^2$$.
Proof of the theorem:
Given:

A triangle right angled at $$B$$.

That is $$\angle ABC$$ $$=$$ $$90^{\circ}$$.

To prove:

$$AC^2=AB^2+BC^2$$

Construction:

Construct a line from $$B$$ to $$AC$$ to intersect at $$D$$ such that $$BD \perp AC$$.

Proof:

Consider the triangles $$ABC$$ and $$BDC$$.
If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, then triangles on both sides of  the perpendicular are similar to the whole triangle and to each other.
By the theorem, we have $$\Delta ABC$$ $$\sim$$ $$\Delta BDC$$.

Hence, the ratio of the corresponding sides of the triangles are equal.

That is, $$\frac{BC}{CD} = \frac{AC}{BC}$$.

This implies, $$BC^{2} = AC \times CD$$        ……$$(1)$$

Now consider the triangles $$ABC$$ and $$ABD$$.

Similarly, by the above mentioned theorem we have $$\Delta ABC$$ $$\sim$$ $$\Delta ABD$$.

Hence, the ratio of the corresponding sides of the triangles are equal.

So, $$\frac{AB}{AD} = \frac{AC}{AB}$$.

This implies, $$AB^{2} = AC \times AD$$        ……$$(2)$$

Add equations $$(1)$$ and $$(2)$$ as follows:

$$BC^2 + AB^2$$ $$=$$ $$(AC \times CD) + (AC \times AD)$$

$$=$$ $$AC (CD +AD)$$

$$=$$ $$AC \cdot AC$$

$$=$$ $$AC^2$$.

Therefore, $$AC^2 = AB^2 + BC^2$$.

Hence, the proof.
Example:
In a right angled triangle, if the measure of the hypotenuse is $$29 cm$$ and one of its sides is $$21$$ $$cm$$ then, find the length of the other side.

Solution:

Let the triangle be $$ABC$$ right angled at $$B$$.

This implies that the side $$AC$$ is the hypotenuse.

By the Pythagorean theorem, we have $$AC^2 = AB^2 + BC^2$$.

Thus, $$AB^2 = AC^2 - BC^2$$.

$$\Rightarrow AB^2 = 29^2 -21^2$$

$$= 841 - 441$$

$$= 400$$

Hence, $$AB = \sqrt{400}$$.

$$AB = 20$$

Therefore, the length of the other side is $$20$$ $$cm$$.