Question

Abcd is a square of side 4cm. a point p is on ad find the minimum possible lenght of bp and cp together (in cm).

Answer 1

THE ONLY IMPORTANT POINT is : The minimum value of BP+CP is when the point P is the mid-point. Now calculate BP+CP taking P as mid point.  answer will be 4√5.

Answer 2

Answer:

Minimum possible length of BP and Cp together is 4√5 = 8.9 cm

Step-by-step explanation:

Given: Side of Square ABCD = 4 cm

           Pont P on Side AD

To find: PB + CP

Case 1: when P is mid point of AD

Draw PX ⊥ BC

now, in Δ PBX

by Pythagoras theorem,

PB² = PX² + BX²

PB² = 4² + 2² ( PX = AB = 4 cm and BX = $\frac{4}{2}$ = 2 cm )

PB² = 16 +4

PB² = 20

PB = √20

PB = 2√5 cm

Also, CP = 2√5 cm

⇒ BP + CP = 2√5 + 2√5 = 4√5 cm = 8.9 cm (approx.)

Case 2: when Point P is 1 cm away from point A

Draw PY ⊥ BC

now, in Δ PBY

by Pythagoras theorem,

PB² = PY² + BY²

PB² = 4² + 1² ( PY = AB = 4 cm and BY = AP = 1 cm )

PB² = 16 + 1

PB² = 17

PB = √17

PB = √17 cm

in Δ PCY

by Pythagoras theorem,

PC² = PY² + CY²

PC² = 4² + 3² ( PY = AB = 4 cm and CY = CB - BY = 4 - 1 = 3 cm )

PC² = 16 + 9

PC² = 25

PC = √25

PC = 5 cm

⇒ BP + CP = √17 + 5 = 5√17 cm = 9.1 cm (approx.)

Therefore, Minimum possible length of BP and Cp together is 4√5 = 8.9 cm