Question

1. find the length of the missing side of each triangle. 4 mi 3 mi 15 yd 13 yd pythagorean theorem: the sum of the squares of the legs is equal to the square of the hypotenuse. formula: a² + b² = c² the legs are represents with variables a and b the hypotenuse is represented with the variable c 7. rhombus abcd has vertices whose coordinates are a(1, 2), b(4, 6), c(7, 2), and d(4, -2). determine and state the perimeter of rhombus abcd. solving for perimeter on a coordinate plane: find the distance of each segment by creating a right triangle for each segment. then use the pythagorean theorem to solve for the distance. to solve for the perimeter you must add up all the sides.

Explanation:

Step1: Find the missing side of the first triangle

Use the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $a = 3$ mi and $b = 4$ mi.
$x=\sqrt{3^{2}+4^{2}}=\sqrt{9 + 16}=\sqrt{25}=5$ mi

Step2: Find the missing side of the second triangle

Here, the hypotenuse is 15 yd and one leg is 13 yd. Let the missing leg be $x$. Then $x=\sqrt{15^{2}-13^{2}}=\sqrt{(15 + 13)(15 - 13)}=\sqrt{28\times2}=\sqrt{56}=2\sqrt{14}$ yd

Step3: Find the side - length of the rhombus

Use the distance formula $d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}$ for two points $(x_1,y_1)$ and $(x_2,y_2)$. For side AB with $A(1,2)$ and $B(4,6)$:
$AB=\sqrt{(4 - 1)^{2}+(6 - 2)^{2}}=\sqrt{3^{2}+4^{2}}=\sqrt{9+16}=5$
Since a rhombus has all sides equal, the perimeter of rhombus $ABCD$ is $4\times5 = 20$

Answer:

The missing side of the first triangle is 5 mi, the missing side of the second triangle is $2\sqrt{14}$ yd, and the perimeter of rhombus $ABCD$ is 20.