The perimeter of an equilateral triangle measure 48 inches. The midpoints of the sides of the triangle are joined to form another equilateral triangle with sides that are half the length of the outer triangle. This process is continued until three smaller triangles are created within the first triangle. Use a geometric series to model the perimeters of the four triangles.

The sum of the perimeters of all four triangles is
inches.

Step-by-step explanation:

Side of the first triangle = 48/3 = 16 in

Side of the inner triangles, common ratio is 1/2:

16, 8, 4, 2

Perimeter of the triangles:

48, 24, 12, 6

Sum of perimeters:

here

step-by-step explanation:

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12