Question

question 16 (5 points)
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whats the exact area and the exact perimeter of square efgh shown in the coordinate plane?
a) area = 20√2 square units; perimeter = 50 units
b) area = 50 square units; perimeter = 20√2 units
c) area = 50 units; perimeter = 20√2 square units
d) area = 20√2 units; perimeter = 50 square units

Explanation:

Step1: Find the length of the diagonals

From the coordinate grid, we can see that the length of diagonal \( EG \) (horizontal diagonal) is 10 units (since it spans from -5 to 5 on the x - axis, \( 5 - (-5)=10 \)) and the length of diagonal \( FH \) (vertical diagonal) is 10 units (spans from -5 to 5 on the y - axis, \( 5 - (-5) = 10\)).

Step2: Calculate the area of the square

For a square (which is a rhombus) with diagonals \( d_1 \) and \( d_2 \), the area formula is \( A=\frac{1}{2}\times d_1\times d_2 \).
Substituting \( d_1 = 10 \) and \( d_2=10 \) into the formula, we get \( A=\frac{1}{2}\times10\times10 = 50 \) square units.

Step3: Calculate the side length of the square

Using the Pythagorean theorem, if the side length of the square is \( s \), and the diagonals of the square bisect each other at right angles, then the side length \( s \) can be found from the right triangle formed by half of each diagonal. Half of each diagonal is \( \frac{10}{2}=5 \) units.
By Pythagorean theorem \( s=\sqrt{5^{2}+5^{2}}=\sqrt{25 + 25}=\sqrt{50}=5\sqrt{2} \) units.

Step4: Calculate the perimeter of the square

The perimeter of a square \( P = 4s \). Substituting \( s = 5\sqrt{2} \), we get \( P=4\times5\sqrt{2}=20\sqrt{2} \) units.

Answer:

B) Area = 50 square units; perimeter = \( 20\sqrt{2} \) units