Question

Observe the right triangle ABC, right angled at B as shown below.
What is the length of PC?

(a) 2.5 cm
(b) 4.5 cm
(d) 7.5 cm
(c) 6 cm

Please answer this question.​

Answer 1

(d) 7.5 cm

From th question the correct option is d.

The length of  the $PC$ is $7.5$

As shown in the given image triangle ABC is given with a right angle B.

In a triangle ABC :

$AB$ = $5cm$

$AP$= $x$

$PC$ = $x+5cm$

So the  length of  the $PC$ is $7.5$

Answer 2

Given  : right triangle ABC, right angled at B

To Find : Length of PC

(a) 2.5 cm

(b) 4.5 cm

(d) 7.5 cm

(c) 6 cm

Solution:

in ΔAPB using Pythagorean theorem :

AB² = AP² + BP²

=> 5² =  x² +  BP²

=> BP² = 25 - x²

Using Geometric mean (Altitude) Theorem

The length of the altitude from the right angle to the hypotenuse in a right triangle is the geometric mean of the lengths of segments the altitude divides the hypotenuse into.

BP² = AP .CP

=> 25 - x² = x(x +5)

=> 25 - x² = x² +5x

=> 2x² + 5x - 25 = 0

=> 2x² + 10x - 5x - 25 = 0

=> 2x(x + 5) - 5(x +  5) =  0

=> (2x - 5)(x + 5) = 0

=> x = 5/2  , x = - 5

Length can not be negative

Hence x = 5/2

Length of PC  =5 + x  = 5 + 5/2  = 7.5  cm

Hence correct option is  (d) 7.5 cm

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