Question

1. the vertices of rectangle abcd are a(-10, 7), b(6, 7), c(6, -5), and d(-10, -5). what is the perimeter, area, and diagonal length of the rectangle? show your work on the back of this paper. round your answer to the nearest hundredth.
2. how many open dots are in the 8th term of the following sequence? whats the pattern? show your work or explain how you found your solution.

term 1: o
term 2: o
oo
ooo
term 3: o
oo
ooo
oooo
ooooo
term 4: o
oo
ooo
oooo
ooooo
oooooo
ooooooo

Explanation:

Step1: Find length and width of rectangle

Use distance formula for two - points $(x_1,y_1)$ and $(x_2,y_2)$ which is $d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. For length, between $A(-10,7)$ and $B(6,7)$: $l=\sqrt{(6-( - 10))^2+(7 - 7)^2}=\sqrt{(16)^2+0^2}=16$. For width, between $B(6,7)$ and $C(6,-5)$: $w=\sqrt{(6 - 6)^2+(-5 - 7)^2}=\sqrt{0+( - 12)^2}=12$.

Step2: Calculate perimeter of rectangle

Perimeter formula of rectangle is $P = 2(l + w)$. Substitute $l = 16$ and $w = 12$: $P=2(16 + 12)=2\times28 = 56$.

Step3: Calculate area of rectangle

Area formula of rectangle is $A=l\times w$. Substitute $l = 16$ and $w = 12$: $A=16\times12 = 192$.

Step4: Calculate diagonal length of rectangle

Use Pythagorean theorem $d=\sqrt{l^{2}+w^{2}}$. Substitute $l = 16$ and $w = 12$: $d=\sqrt{16^{2}+12^{2}}=\sqrt{256 + 144}=\sqrt{400}=20$.

Step5: Analyze dot - pattern sequence

In the given sequence, the number of dots in each term forms an arithmetic - like pattern for the number of rows and columns of dots. The number of dots in the $n$th term can be found by the sum of the first $n$ positive integers. The formula for the sum of the first $n$ positive integers is $S_n=\frac{n(n + 1)}{2}$. For $n = 8$, $S_8=\frac{8\times(8 + 1)}{2}=\frac{8\times9}{2}=36$.

Answer:

Perimeter of rectangle: $56$
Area of rectangle: $192$
Diagonal length of rectangle: $20$
Number of open - dots in 8th term of sequence: $36$