Given: △AMF AM=MF=AF=a, KLPS is a square Find: Area of KLPS

Question

Given: △AMF

      AM=MF=AF=a, KLPS is a square

    Find: Area of KLPS

Answer 1

let PS=x and KS, LK and LP are all "x"

triangle MAF is isosceles so angle MFA is 60°

tan60°=√3

tan=opposite/adjacent

tan60°=x/SF

√3=x/SF

√3 can be written as 3/√3

3/√3=x/SF

cross multiply

3SF=√3x

SF=√3x/3

AK=√3x/3 also

KS=SP=x

KS=a-AK-SF

KS=a-2√3x/3

but KS=x

so, x=a-2√3x/3

a=x+(2√3/3)x

a=x(1+2√3/3)

x=a/(1+2√3/3)

x=a/([3+2√3]/3

x=3a/(3+2√3)

the area of the square is x^2

x^2=9a²/(9+12√3+12)

x^2=9a²/(21+12√3)

x^2=3a²/(7+4√3) is the area of the square KLPS

Answer 2

Hi Hunter!

We know KLPS is a square, then all sides are equal and all the internal angles are 90°. Let's call the measure of each side as "b".

We also know that AMF is a equilateral triangle, then all the internal angles are 60°.

Looking at the triangle PSF, from the tangent of 60°, we have:

b/SF = √3

SF = (b√3)/3

For the same reason AK = (b√3)/3

We know AF = a, then:

AK + KS + SF = (b√3)/3 + b + (b√3)/3 = a

b(3+ 2√3) = 3a

b(3+ 2√3)*(3- 2√3) = 3a*(3- 2√3)

b*(-2) = 3a*(3- 2√3)

b = 3a/2 * ( 2√3 - 3)

The area of KLPS is b²

b² = 9a²/4 * (12 -12√3 + 9) = 27a²/4 * (7-4√3)

Answer 3

Let s = side of the square

ΔMLP is equilateral (because it is equiangular

triangle AKL is a 30-60 right Δ

length of AL = (2/3)s√3

length of LM = s

s+(2/3)s√3=a

s = 3a/(3+2√3) and the area of the square is s²

Answer 4

AL^2-AK^2=KL^2