Question

abc is a right triangle right angled at c let bc =a ca =b ab =.c and p be the length of perpendicular from c to.ab prove that cp=ab​

Answer 1

Given :-

A right triangle ABC right angled at C such that

• AB =c

• BC = a

• CA = b

and

• Length of perpendicular from C to AB is p.

To Prove :-

• pc = ab

Formula Used :-

$\boxed{\bf\:Area_{(triangle)} = \dfrac{1}{2} \times base \times height}$

Solution :-

Given that,

In right triangle ABC,

• AB = c

• BC = a

• CA = b

• CD = p

Now,

Area of triangle ABC with base BC = a and height CA = b is given by

$\boxed{\bf\:Area_{( \triangle \: ABC)} = \dfrac{1}{2} \times a\times b} - - - (1)$

Again,

Area of triangle ABC with base AB = c and height CD = p is given by

$\boxed{\bf\:Area_{( \triangle \: ABC)} = \dfrac{1}{2} \times c\times p} - - - (2)$

From equation (1) and equation (2), we concluded that

$\rm :\longmapsto\:\cancel{\dfrac{1}{2}} \times a \times b = \cancel{\dfrac{1}{2}} \times p \times c$

$\bf\implies \:ab = pc$

${\boxed{\boxed{\bf{Hence, Proved}}}}$

Additional Information :-

1. Sum of all interior angles of a triangle is 180°.

2. Exterior angle of a triangle is always equal to sum of the interior opposite angles.

3. Side opposite to greater angle is always greater.

4. Angle opposite to longest side is always greater.

5. Sum of two sides of a triangle is greater than third side.