A square and an equilateral triangle part 2

Question

The length of a side of the square is x cm. 

The length of a side of the equilateral triangle is 2 cm more than the length of a side of the square.

The perimeter of the square is equal to the perimeter of the equilateral triangle.

(Already answered, the perimeter of the square is 24cm)

 

The length of the diagonal of the square is y cm.

The height of the equilateral triangle is z cm.

Which has the greater value, y or z?

 

Answer 1

Perim of square = 4x and perim of triangle is 3(x+2). Equating these two perimeters gives x = 6.

 

Diagonal of square is y = x√2 which is 6√2 or about 8.48528

 

z= 4√4 or about 6.92820, using well known properties of all equilateral triangles.

 

Now you know y > z

Answer 2

the square has side lengths of 6 cm

the equilateral triangle has side lengths of 8 cm

the diagonal y and the two side lengths of 6 make up a right triangle

y^2-6^2+6^2

y^2=36+36

y^2=72

y=√72

y=√(36*2)

y=6√2=6*1.4142=8.4852 cm

 

the height h of the equilateral triangle, one of the sides, and half of another side make up a right triangle

z^2+4^2=8^2  (half of one side is half of 8)

z^2+16=64

z^2=64-16

z^2=48

z=√48

z=√(16*3)

z=4√3

z=4*1.732

z=6.928 cm

y has the greater value