Question

∆ABC is an equilateral triangle point P is on base BC such that PC =1/3BC,if AB=6 find AP

Answer 1

Hello Dear.

Refers to the attachment for the Question.

Construction ⇒ Draw the Perpendicular from the point A to the Base BC.

Solution ⇒
∵ All the sides of the Equilateral Triangles are equal,
∴ AB = BC = AC = 6 units.

Also,
PC = 1/3 BC
= 1/3 × 6
= 2 units.

We know Perpendicular in the Equilateral triangles bisects the opposite sides,
∴ QB = QC.
    = 1/2 × BC.
    = 1/2 × 6
    = 3 units.

In Δ AQB,
Applying Pythagoras Theorem,
AB² = BQ² + AQ² 
⇒ (6)² = (3)² + AQ²
⇒ AQ² = 36 - 9
⇒ AQ² = 27
⇒ AQ = 3√3 units.

In ΔAQP,
PQ = QC - PC
= 3 - 2
= 1 units.

Applying Pythagoras Theorem,
AP² = PQ² + AQ²
⇒ AP² = (1)² + (3√3)²
⇒ AP² = 1 + 27
⇒ AP = √28
⇒ AP = 2√7 units.


Hence, the length of the AP is 2√7 units.



Hope it helps.

Answer 2

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