Question

solve this...................................​

Answer 1

Answer:

  perimeter = 3√(25 + 12√3) ≈ 20.3

  area = (36 + 25√3)/4  ≈  19.8

Step-by-step explanation:

First find the length of the side of the equilateral triangle ABC.

Rotate 60° anticlockwise around B, sending P to Q.  Then BPQ is an equilateral triangle with side length 4.  Also, as AQ is the image of CP which has length 5, the triangle APQ is a 3:4:5 triangle, so it has a right angle at P.

Let D be the foot of the perpendicular from B to the line AP extended.  So triangle ABD has a right angle at D, and what we need to work out is the length of the hypotenuse AB as this is the side of the original equilateral triangle ABC.

Since BPQ is equilateral, BD is half of PQ and DP is √3 times BD.  So BD=2 and DP=2√3.  Thus AD = AP+DP=3+2√3.  Now by Pythagoras' Theorem:

 AB² = BD² + AD² = 2² + (3+2√3)² = 4 + 9 + 12 + 12√3 = 25 + 12√3

⇒  AB = √(25 + 12√3).

The perimeter of ABC is then

   perimeter = 3×AB = 3√(25 + 12√3) ≈ 20.3

For the area, the height of the equilateral triangle is √3/2 × AB, so

   area = 1/2 × AB × √3/2 × AB = √3/4 × AB² =  (36 + 25√3)/4  ≈  19.8

Answer 2

hope it helps you

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