Recall the following properties of a rhombus, as shown by the figure above: The diagonals of a rhombus are perpendicular and they bisect each other. Thus, if we know the lengths of the diagonals, we can use the Pythagorean theorem to find the length of a side of the rhombus.

Recall how to find the area of a rhombus:

$\displaystyle \text{Area}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Since we are given the length of one diagonal and the area, we can find the length of the second diagonal.

$\displaystyle \text{diagonal 2}=\frac{2(\text{Area})}{\text{diagonal 1}}$

Plug in the given values to find the length of the second diagonal.

$\displaystyle \text{diagonal 2}=\frac{2(102)}{12}=17$

Now, notice that the halves of each diagonal make up a right triangle that has the side length of the rhombus as its hypotenuse.

$\displaystyle \text{Half diagonal 1}=\frac{12}{2}=6$

$\displaystyle \text{Half diagonal 2}=\frac{17}{2}=8.5$

Now, use these half values in the Pythagorean Theorem to find the length of the side of a rhombus.

$\displaystyle \text{Side of Rhombus}=\sqrt{6^2+8.5^2}=\sqrt{108.25}$

Finally, recall that a rhombus has four equal side lengths. To find the perimeter, multiply the length of a side by four.

$\displaystyle \text{Perimeter of rhombus}=4\sqrt{108.25}=41.62$

Make sure to round to $\displaystyle 2$ places after the decimal.

Recall the following properties of a rhombus, as shown by the figure above: The diagonals of a rhombus are perpendicular and they bisect each other. Thus, if we know the lengths of the diagonals, we can use the Pythagorean theorem to find the length of a side of the rhombus.

Recall how to find the area of a rhombus:

$\displaystyle \text{Area}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Since we are given the length of one diagonal and the area, we can find the length of the second diagonal.

$\displaystyle \text{diagonal 2}=\frac{2(\text{Area})}{\text{diagonal 1}}$

Plug in the given values to find the length of the second diagonal.

$\displaystyle \text{diagonal 2}=\frac{2(150)}{15}=20$

Now, notice that the halves of each diagonal make up a right triangle that has the side length of the rhombus as its hypotenuse.

$\displaystyle \text{Half diagonal 1}=\frac{15}{2}=7.5$

$\displaystyle \text{Half diagonal 2}=\frac{20}{2}=10$

Now, use these half values in the Pythagorean Theorem to find the length of the side of a rhombus.

$\displaystyle \text{Side of Rhombus}=\sqrt{7.5^2+100^2}=\sqrt{156.25}$

Finally, recall that a rhombus has four equal side lengths. To find the perimeter, multiply the length of a side by four.

$\displaystyle \text{Perimeter of rhombus}=4\sqrt{156.25}=50$

Recall the following properties of a rhombus, as shown by the figure above: The diagonals of a rhombus are perpendicular and they bisect each other. Thus, if we know the lengths of the diagonals, we can use the Pythagorean theorem to find the length of a side of the rhombus.

Recall how to find the area of a rhombus:

$\displaystyle \text{Area}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Since we are given the length of one diagonal and the area, we can find the length of the second diagonal.

$\displaystyle \text{diagonal 2}=\frac{2(\text{Area})}{\text{diagonal 1}}$

Plug in the given values to find the length of the second diagonal.

$\displaystyle \text{diagonal 2}=\frac{2(48)}{8}=12$

Now, notice that the halves of each diagonal make up a right triangle that has the side length of the rhombus as its hypotenuse.

$\displaystyle \text{Half diagonal 1}=\frac{8}{2}=4$

$\displaystyle \text{Half diagonal 2}=\frac{12}{2}=6$

Now, use these half values in the Pythagorean Theorem to find the length of the side of a rhombus.

$\displaystyle \text{Side of Rhombus}=\sqrt{4^2+6^2}=\sqrt{52}$

Finally, recall that a rhombus has four equal side lengths. To find the perimeter, multiply the length of a side by four.

$\displaystyle \text{Perimeter of rhombus}=4\sqrt{52}=28.84$

Make sure to round to $\displaystyle 2$ places after the decimal.

Recall the following properties of a rhombus, as shown by the figure above: The diagonals of a rhombus are perpendicular and they bisect each other. Thus, if we know the lengths of the diagonals, we can use the Pythagorean theorem to find the length of a side of the rhombus.

Recall how to find the area of a rhombus:

$\displaystyle \text{Area}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Since we are given the length of one diagonal and the area, we can find the length of the second diagonal.

$\displaystyle \text{diagonal 2}=\frac{2(\text{Area})}{\text{diagonal 1}}$

Plug in the given values to find the length of the second diagonal.

$\displaystyle \text{diagonal 2}=\frac{2(152)}{16}=19$

Now, notice that the halves of each diagonal make up a right triangle that has the side length of the rhombus as its hypotenuse.

$\displaystyle \text{Half diagonal 1}=\frac{16}{2}=8$

$\displaystyle \text{Half diagonal 2}=\frac{19}{2}=9.5$

Now, use these half values in the Pythagorean Theorem to find the length of the side of a rhombus.

$\displaystyle \text{Side of Rhombus}=\sqrt{8^2+9.5^2}=\sqrt{154.25}$

Finally, recall that a rhombus has four equal side lengths. To find the perimeter, multiply the length of a side by four.

$\displaystyle \text{Perimeter of rhombus}=4\sqrt{154.25}=49.68$

Make sure to round to $\displaystyle 2$ places after the decimal.

Recall the following properties of a rhombus, as shown by the figure above: The diagonals of a rhombus are perpendicular and they bisect each other. Thus, if we know the lengths of the diagonals, we can use the Pythagorean theorem to find the length of a side of the rhombus.

Recall how to find the area of a rhombus:

$\displaystyle \text{Area}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Since we are given the length of one diagonal and the area, we can find the length of the second diagonal.

$\displaystyle \text{diagonal 2}=\frac{2(\text{Area})}{\text{diagonal 1}}$

Plug in the given values to find the length of the second diagonal.

$\displaystyle \text{diagonal 2}=\frac{2(187)}{17}=22$

Now, notice that the halves of each diagonal make up a right triangle that has the side length of the rhombus as its hypotenuse.

$\displaystyle \text{Half diagonal 1}=\frac{17}{2}=8.5$

$\displaystyle \text{Half diagonal 2}=\frac{22}{2}=11$

Now, use these half values in the Pythagorean Theorem to find the length of the side of a rhombus.

$\displaystyle \text{Side of Rhombus}=\sqrt{8.5^2+11^2}=\sqrt{193.25}$

Finally, recall that a rhombus has four equal side lengths. To find the perimeter, multiply the length of a side by four.

$\displaystyle \text{Perimeter of rhombus}=4\sqrt{193.25}=55.61$

Make sure to round to $\displaystyle 2$ places after the decimal.

Recall the following properties of a rhombus, as shown by the figure above: The diagonals of a rhombus are perpendicular and they bisect each other. Thus, if we know the lengths of the diagonals, we can use the Pythagorean theorem to find the length of a side of the rhombus.

Recall how to find the area of a rhombus:

$\displaystyle \text{Area}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Since we are given the length of one diagonal and the area, we can find the length of the second diagonal.

$\displaystyle \text{diagonal 2}=\frac{2(\text{Area})}{\text{diagonal 1}}$

Plug in the given values to find the length of the second diagonal.

$\displaystyle \text{diagonal 2}=\frac{2(228)}{19}=24$

Now, notice that the halves of each diagonal make up a right triangle that has the side length of the rhombus as its hypotenuse.

$\displaystyle \text{Half diagonal 1}=\frac{19}{2}=9.5$

$\displaystyle \text{Half diagonal 2}=\frac{24}{2}=12$

Now, use these half values in the Pythagorean Theorem to find the length of the side of a rhombus.

$\displaystyle \text{Side of Rhombus}=\sqrt{9.5^2+12^2}=\sqrt{234.25}$

Finally, recall that a rhombus has four equal side lengths. To find the perimeter, multiply the length of a side by four.

$\displaystyle \text{Perimeter of rhombus}=4\sqrt{234.25}=61.22$

Make sure to round to $\displaystyle 2$ places after the decimal.

Recall the following properties of a rhombus, as shown by the figure above: The diagonals of a rhombus are perpendicular and they bisect each other. Thus, if we know the lengths of the diagonals, we can use the Pythagorean theorem to find the length of a side of the rhombus.

Recall how to find the area of a rhombus:

$\displaystyle \text{Area}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Since we are given the length of one diagonal and the area, we can find the length of the second diagonal.

$\displaystyle \text{diagonal 2}=\frac{2(\text{Area})}{\text{diagonal 1}}$

Plug in the given values to find the length of the second diagonal.

$\displaystyle \text{diagonal 2}=\frac{2(96)}{12}=16$

Now, notice that the halves of each diagonal make up a right triangle that has the side length of the rhombus as its hypotenuse.

$\displaystyle \text{Half diagonal 1}=\frac{12}{2}=6$

$\displaystyle \text{Half diagonal 2}=\frac{16}{2}=8$

Now, use these half values in the Pythagorean Theorem to find the length of the side of a rhombus.

$\displaystyle \text{Side of Rhombus}=\sqrt{6^2+8^2}=\sqrt{100}=10$

Finally, recall that a rhombus has four equal side lengths. To find the perimeter, multiply the length of a side by four.

$\displaystyle \text{Perimeter of rhombus}=4(10)=40$

Recall the following properties of a rhombus, as shown by the figure above: The diagonals of a rhombus are perpendicular and they bisect each other. Thus, if we know the lengths of the diagonals, we can use the Pythagorean theorem to find the length of a side of the rhombus.

Recall how to find the area of a rhombus:

$\displaystyle \text{Area}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Since we are given the length of one diagonal and the area, we can find the length of the second diagonal.

$\displaystyle \text{diagonal 2}=\frac{2(\text{Area})}{\text{diagonal 1}}$

Plug in the given values to find the length of the second diagonal.

$\displaystyle \text{diagonal 2}=\frac{2(210)}{20}=21$

Now, notice that the halves of each diagonal make up a right triangle that has the side length of the rhombus as its hypotenuse.

$\displaystyle \text{Half diagonal 1}=\frac{20}{2}=10$

$\displaystyle \text{Half diagonal 2}=\frac{21}{2}=10.5$

Now, use these half values in the Pythagorean Theorem to find the length of the side of a rhombus.

$\displaystyle \text{Side of Rhombus}=\sqrt{10^2+10.5^2}=\sqrt{210.25}$

Finally, recall that a rhombus has four equal side lengths. To find the perimeter, multiply the length of a side by four.

$\displaystyle \text{Perimeter of rhombus}=4\sqrt{210.25}=58$

Recall the following properties of a rhombus, as shown by the figure above: The diagonals of a rhombus are perpendicular and they bisect each other. Thus, if we know the lengths of the diagonals, we can use the Pythagorean theorem to find the length of a side of the rhombus.

Recall how to find the area of a rhombus:

$\displaystyle \text{Area}=\frac{(\text{diagonal 1})(\text{diagonal 2})}{2}$

Since we are given the length of one diagonal and the area, we can find the length of the second diagonal.

$\displaystyle \text{diagonal 2}=\frac{2(\text{Area})}{\text{diagonal 1}}$

Plug in the given values to find the length of the second diagonal.

$\displaystyle \text{diagonal 2}=\frac{2(1200)}{40}=60$

Now, notice that the halves of each diagonal make up a right triangle that has the side length of the rhombus as its hypotenuse.

$\displaystyle \text{Half diagonal 1}=\frac{40}{2}=20$

$\displaystyle \text{Half diagonal 2}=\frac{60}{2}=30$

Now, use these half values in the Pythagorean Theorem to find the length of the side of a rhombus.

$\displaystyle \text{Side of Rhombus}=\sqrt{20^2+30^2}=\sqrt{1300}$

Finally, recall that a rhombus has four equal side lengths. To find the perimeter, multiply the length of a side by four.

$\displaystyle \text{Perimeter of rhombus}=4\sqrt{1300}=144.22$

Make sure to round to $\displaystyle 2$ places after the decimal.

The rhombus referenced is below:

A diagonal of a rhombus bisects the angles of the rhombus at its endpoints. Therefore, since $\displaystyle m \angle ABD = 25 ^{\circ }$, it follows that $\displaystyle m \angle DBC = 25 ^{\circ }$ as well. By angle addition,

$\displaystyle m \angle ABC = m \angle ABD + m \angle DBC = 25^{\circ }+ 25^{\circ } = 50^{\circ }$.

As consecutive angles of a rhombus (and, consequently, of a parallelogram), $\displaystyle \angle ABC$ and $\displaystyle \angle BDC$ are supplementary - that is, their measures total $\displaystyle 180^{\circ }$. Therefore,

$\displaystyle m \angle ABC + m \angle BCD = 180^{\circ }$

$\displaystyle 50 ^{\circ } + m \angle BCD = 180^{\circ }$

$\displaystyle 50 ^{\circ } + m \angle BCD - 50 ^{\circ } = 180^{\circ } - 50 ^{\circ }$

$\displaystyle m \angle BCD = 130 ^{\circ }$