The formula to find perimeter of an equilateral triangle is

\(\displaystyle P = 3a\)

where a is the length of one side.  We can multiply that by 3, because an equilateral triangle has 3 equal sides.  Now, to find the length of one side, we will solve for a.

So, we know the perimeter of the equilateral triangle is 27cm.  Knowing this, we can substitute.  We get

\(\displaystyle 27\text{cm} = 3a\)

\(\displaystyle \frac{27\text{cm}}{3} = \frac{3a}{3}\)

\(\displaystyle 9\text{cm} = a\)

\(\displaystyle a = 9\text{cm}\)

Therefore, the length of one side of the equilateral triangle is 9cm.

To find the perimeter of a triangle, we will use the following formula:

\(\displaystyle P = a+b+c\)

where a, b, and c are the lengths of the sides of the triangle.

Now, given the triangle

we can see that it has sides of length 8ft, 4ft, and 7ft.  So, we get

\(\displaystyle P = 8\text{ft} + 4\text{ft} + 7\text{ft}\)

\(\displaystyle P = 19\text{ft}\)

By definition, an equilateral triangle is a triangle with three equal sides. That is, the length of each side of the triangle is going to be the same.

In this problem, we know that the perimeter (the sum of all the lengths) is 42 cm.

Side 1 + Side 2 + Side 3 = 42 cm.

Since the sides are equal, we can write the following equation. \(\displaystyle S =\) one side of the triangle

\(\displaystyle 3S=42 cm.\) 

To find the length of one side of the triangle, we would then divide 3 from both sides.

\(\displaystyle \frac{3}{3}S=\frac{42}{3}\)

\(\displaystyle 1S=14\) Remember that \(\displaystyle 1*S\) is the same as \(\displaystyle S\)

\(\displaystyle S=14\ cm.\)

A key characteristic of a rectangle, parallelogram, and trapezoid is the fact that they each have at least one pair of lines that run parallel to each other. The only option that has lines that may not run parallel to each other is the quadrilateral, which must simply have four sides but has no specifications about parallelism.

While drawing a line across a rectangle (so that it bisects 2 sides) can result in 2 quadrilaterals, squares, or rectangles, a line drawn from one corner to the furthest corner results in two triangles. Therefore, the correct answer choice is 2 triangles.

A line of symmetry is a line that divides a polygon in half and each half is a mirror image of the other. In other words, you can fold the polygon over the symmetry line and each half matches up perfectly. 

For a square there are four lines of symmetry. Two are from the diagonals of the square and two are from connecting the midpoints of the opposite sides.

The distance formula is given by \(\displaystyle d=\sqrt{(x_{2}-x_{1})^{2} + (y_{2}-y_{1})^{2}}\).

Let \(\displaystyle P_{1}=(4,-5)\) and \(\displaystyle P_{2}=(-2,3)\):

\(\displaystyle d=\sqrt{(-2-4)^{2} + (3-(-5)^{2}}=\sqrt{(-6)^{2} + (8)^{2}}=\sqrt{36 + 64}=\sqrt{100}=10\)

The distance formula is given by \(\displaystyle d=\sqrt{(x_{2}-x_{1})^{2} + (y_{2}-y_{1})^{2}}\).

Let \(\displaystyle P_{1}=(1,-3)\) and \(\displaystyle P_{2}=(13,-8)\):

\(\displaystyle d=\sqrt{(13-1)^{2} + (-8-(-3)^{2}}=\sqrt{(12)^{2} + (-5)^{2}}=\sqrt{144 + 25}=\sqrt{169}=13\)

The distance formula is \(\displaystyle d = \sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}\).

Let \(\displaystyle P_{1}= (-2, 1)\) and \(\displaystyle P_{2}=(10, -4)\).

Plug these two points into the distance formula:

\(\displaystyle d = \sqrt{(10-(-2))^{2}+(-4-1)^{2}}=\sqrt{(12)^{2}+(-5)^{2}}=\sqrt{144+25}=\sqrt{169}=13\)