# Triangles For Class 10

As the name suggest, a triangle is a polygon having three sides and three angles. The sum of interior angle of a triangle to be equal to $180^{\circ}$.

Similarity of Triangle-

Two triangles are similar, if

(i) their corresponding angles are equal and

(ii) their corresponding sides are in the same ratio (or proportion).

Pythagoras Theorem-

In a right-angled triangle the sum of squares of two sides is equal to the square of the hypotenuse of the triangle.

Proof of Pythagoras Theorem-

Consider a right triangle, right angled at B.

From Pythagoras Theorem, we have

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

Construction-

Draw $BD \perp AC$

Now, $\bigtriangleup ADC \sim \bigtriangleup ABC$

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

or, $AD. AC = AB^{2}$ ……………(i)

Also, $\bigtriangleup BCDC \sim \bigtriangleup ABC$

So, $\frac{CD}{BC} = \frac{BC}{AC}$

or, $CD. AC = BC^{2}$ ……………(ii)

$AD. AC + CD. AC = AB^{2} + BC^{2}$

$AC(AD + DC) = AB^{2} + BC^{2}$

$AC(AC) = AB^{2} + BC^{2}$

$\Rightarrow AC^{2} = AB^{2} + BC^{2}$<

#### Practise This Question

The ratio of corresponding sides for the pair of triangles whose construction is given as follows :
Triangle ABC of dimesions AB=4cm,BC= 5 cm and ∠B= 60o.
A ray BX is drawn from B making an acute angle with AB.
5 points B1,B2,B3,B4 and B5 are located on the ray such that BB1=B1B2=B2B3=B3B4=B4B5.
B4 is joined to A and a line parallel to B4A is drawn through B5 to intersect the extended line AB at A'.
Another line is drawn through A' parallel to AC, intersecting the extended line BC at C'. Find the ratio of the corresponding sides of ΔABC and ΔABC.