1. Remember that the perimeter is the sum of the lengths of the sides of a figure.To solve this, we are going to use the distance formula: [tex]d= \sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2} [/tex]
where
[tex](x_{1},y_{1})[/tex] are the coordinates of the first point
[tex](x_{2},y_{2})[/tex] are the coordinates of the second point
Length of WZ:
We know form our graph that the coordinates of our first point, W, are (1,0) and the coordinates of the second point, Z, are (4,2). Using the distance formula:
[tex]d_{WZ}= \sqrt{(4-1)^2+(2-0)^2} [/tex]
[tex]d_{WZ}= \sqrt{(3)^2+(2)^2}[/tex]
[tex]d_{WZ}= \sqrt{9+4}[/tex]
[tex]d_{WZ}= \sqrt{13}[/tex]
We know that all the sides of a rhombus have the same length, so
[tex]d_{YZ}= \sqrt{13} [/tex]
[tex]d_{XY}= \sqrt{13} [/tex]
[tex]d_{XW}= \sqrt{13} [/tex]
Now, we just need to add the four sides to get the perimeter of our rhombus:
[tex]perimeter= \sqrt{13} + \sqrt{13} + \sqrt{13} + \sqrt{13} [/tex]
[tex]perimeter=4 \sqrt{13} [/tex]
We can conclude that the perimeter of our rhombus is [tex]4 \sqrt{13} [/tex] square units.
2. To solve this, we are going to use the arc length formula: [tex]s=r \alpha [/tex]
where
[tex]s[/tex] is the length of the arc.
[tex]r[/tex] is the radius of the circle.
[tex] \alpha [/tex] is the central angle in radians
We know form our problem that the length of arc PQ is [tex] \frac{8}{3} \pi [/tex] inches, so [tex]s=\frac{8}{3} \pi[/tex], and we can infer from our picture that [tex]r=15[/tex]. Lest replace the values in our formula to find the central angle POQ:
[tex]s=r \alpha [/tex]
[tex]\frac{8}{3} \pi=15 \alpha [/tex]
[tex] \alpha = \frac{\frac{8}{3} \pi}{15} [/tex]
[tex] \alpha = \frac{8}{45} \pi [/tex]
Since [tex] \alpha =POQ[/tex], We can conclude that the measure of the central angle POQ is [tex]\frac{8}{45} \pi[/tex]
3. A cross section is the shape you get when you make a cut thought a 3 dimensional figure. A rectangular cross section is a cross section in the shape of a rectangle. To get a rectangular cross section of a particular 3 dimensional figure, you need to cut in an specific way. For example, a rectangular pyramid cut by a plane parallel to its base, will always give us a rectangular cross section.
We can conclude that the draw of our cross section is:
