Question

Observe the right triangle ABC, right angled at B as shown below.
what is the length of pc​

Answer 1

Given:

Here a triangle ABC is given with a right angle B as shown in the given picture.

• A perpendicular is drawn from B to AC which cuts AC on P.
• AB= 5cm , PC= $x$+5 cm, AP=$x$cm

To find:

We have to find the length of PC.

Solution:

• To find the length of PC, first we have to find the value of $x$.
• In ΔABC and ΔABP,
• ∠ABC and ∠APB are right angle, and BP is common side.
• So, ΔABC≈ΔABP
• Now, AC= AP+PC= x+x+5 cm
• So, from the theory of homogeneity, we can write,
• $\frac{AC}{AB}=\frac{AB}{AP} \\or,\frac{x+x+5}{5}=\frac{5}{x}\\or,2x^{2} +5x=25\\or, 2x^{2} +5x-25=0\\or, 2x^{2} +10x-5x-25=0\\or,2x(x+5)-5(x+5)=0\\or,(x+5)(2x-5)=0$
• so, $2x-5=0 or, x=\frac{5}{2}$
• So, PC= $x+5=\frac{5}{2} +5=7.5$cm

Final answer:

The length of PC is 7.5cm [option c].

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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