MathematicsIs it true that cp=ab , if △ABC is a right triangle in which ∠C= 90 ° and CD⊥AB and BC=a,CA=b,AB=c and CD=p ?

Is it true that $c p = a b$ , if $△ A B C$ is a right triangle in which $∠ C = 90 °$ and $C D ⊥ A B$ and $B C = a, C A = b, A B = c$ and $C D = p$ ?

We are given that

∠ C = 90 °, C D ⊥ A B, B C = a, C A = b, A B = c, C D = p

and we have to check if

c p = a b

is true or not.

The area of a triangle is

12 × b a s e × h e i g h t

.
First, we will find the area

Δ A B C

with base AB,

⇒ Δ A B C = 12 A B × C D ⇒ 12 × c × p ⇒ 12 c p

Then we will find the area of

Δ A B C

with base BC,

⇒ Δ A B C = 12 B C × A C ⇒ 12 × a × b ⇒ 12 a b

Thus,

⇒ 12 c p = 12 a b ⇒ c p = a b

Hence, the given statement is true and option 1 is correct.

JEE