the endpoints of one diagonal of a rhombus are (0, -8) and (8, -4). if the coordinates of the 3rd vertex are…

Explanation:

Step1: Recall mid - point property of rhombus

In a rhombus, the diagonals bisect each other. Let the endpoints of the diagonal be $A(0,-8)$ and $B(8, - 4)$, and the third vertex be $C(1,0)$. Let the fourth vertex be $D(x,y)$. The mid - point of the diagonal with endpoints $A$ and $B$ is the same as the mid - point of the diagonal with endpoints $C$ and $D$. The mid - point formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $M=(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. The mid - point of $AB$ is $(\frac{0 + 8}{2},\frac{-8-4}{2})=(4,-6)$.

Step2: Use mid - point formula for $CD$

The mid - point of $CD$ where $C(1,0)$ and $D(x,y)$ is $(\frac{1 + x}{2},\frac{0 + y}{2})$. Since the mid - points of the two diagonals are equal, we have the following two equations: $\frac{1 + x}{2}=4$ and $\frac{y+0}{2}=-6$.

Step3: Solve for $x$ and $y$

From $\frac{1 + x}{2}=4$, we multiply both sides by 2 to get $1 + x = 8$, then $x=7$. From $\frac{y + 0}{2}=-6$, we multiply both sides by 2 to get $y=-12$.

Step4: Recall density - mass - volume formula

The formula for mass $m$, density $\rho$ and volume $V$ is $m=\rho V$. We are given $m = 3.75$ kg. First, convert the mass to grams: $m=3.75\times1000 = 3750$ g and $\rho=7.8$ g/cm³.

Step5: Solve for volume

From $m=\rho V$, we can express $V=\frac{m}{\rho}$. Substitute $m = 3750$ g and $\rho=7.8$ g/cm³ into the formula: $V=\frac{3750}{7.8}\approx480.8$ cm³.

Answer:

1. A. $(7,-12)$
2. 480.8 cubic centimeters