Question

fill - in - the - blank text item interaction
find the area of polygon stuv below. each unit on the graph measures 1 centimeter. round
your answer to the nearest tenth, if necessary.
the area of polygon stuv is
cm².

Explanation:

Step1: Identify the diagonals of the rhombus

The diagonals of the rhombus \(STUV\) can be found using the distance formula (or by counting units on the coordinate - grid).
The length of diagonal \(d_1\) (from \(S(2,6)\) to \(U(2, - 1)\)): \(d_1=\vert6-(-1)\vert=\vert6 + 1\vert=7\) cm.
The length of diagonal \(d_2\) (from \(V(-2,2)\) to \(T(5,3)\)):
Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), here \(x_1=-2,y_1 = 2,x_2=5,y_2 = 3\)
\(d_2=\sqrt{(5+2)^2+(3 - 2)^2}=\sqrt{49 + 1}=\sqrt{50}=5\sqrt{2}\approx7.1\) cm.
Another way (by counting the number of units in the right - triangle formed by the difference in \(x\) and \(y\) coordinates):
The horizontal change \(\Delta x=5-(-2)=7\) and the vertical change \(\Delta y=3 - 2 = 1\). Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), \(d_2=\sqrt{7^{2}+1^{2}}=\sqrt{49 + 1}=\sqrt{50}\approx7.1\) cm.

Step2: Use the formula for the area of a rhombus

The formula for the area of a rhombus is \(A=\frac{1}{2}d_1d_2\).
Substitute \(d_1 = 7\) and \(d_2=\sqrt{50}\approx7.1\) into the formula:
\(A=\frac{1}{2}\times7\times\sqrt{50}\)
\(A=\frac{7\sqrt{50}}{2}=\frac{7\times5\sqrt{2}}{2}=\frac{35\sqrt{2}}{2}\approx\frac{35\times1.414}{2}\)
\(A=\frac{49.49}{2}=24.745\approx24.7\) \(cm^{2}\)

Another approach:
We can also use the method of dividing the polygon into triangles.
The polygon \(STUV\) can be divided into two congruent triangles.
For example, triangle \(SVU\) and triangle \(STU\).
The area of a triangle is \(A=\frac{1}{2}bh\).
If we consider the base and height with respect to the coordinate - grid.
The area of the polygon \(STUV\) can be calculated as the sum of the areas of four right - angled triangles.
Let's use the formula for the area of a rhombus \(A=\frac{1}{2}d_1d_2\) where \(d_1\) is the vertical distance between \(S\) and \(U\) (\(d_1 = 7\)) and \(d_2\) is the distance between \(V\) and \(T\) (calculated as above \(\approx7.1\))

Answer:

\(24.7\)