a Recall that the area of a square is equal to its squared side length.
B
b Find the side lengths of squares ABCD and WXYZ using the formula for the area of square.
C
c If the area of a square is g square meters, the side length of this square is sqrt(g) meters.
A
a 153 cm^2
B
b DZ=5 cm
C
c 9/4g square meters
Practice makes perfect
a Let's begin with recalling that the area of a square is equal to its squared side length. Since we know that the area of a square ABCD is 81 cm^2, we can use this information to evaluate side lengths of this square. Let's call each side length of this square a.
Each side of the square ABCD is equal to 9 cm. We are also given that each side of square ABCD is extended by sides of equal length, that is 3 cm. Let's add this information to the given picture.
Since each angle in a square is a right angle, ∠ZCY is also a right angle as it creates a pair of supplementary angles with the angle of a square. Therefore, triangle ZCY is a right triangle.
Next, we can find the side length of a square WXYZ using the Pythagorean Theorem. Let's recall that the sum of squared legs of a right triangle is equal to its squared hypotenuse. In △ZCY, the legs are 3+ 9=12 and 3. Let w be the side of square WXYZ.
As we recalled at the beginning, the area of a square is equal to its squared side length. Therefore, the area of WXYZ is equal to 153 cm^2.
b Again, let's recall that the area of a square is equal to its squared side length. Since we are given the areas of both squares, we should calculate the square roots of these areas to find the side length of each square.
Square
Area
Side length
ABCD
49
sqrt(49)= 7
WXYZ
169
sqrt(169)= 13
Now, let's look at the given picture. Let x be the length of sides that extend the sides of ABCD. Recall that all angles in a square are right angles. Therefore, ∠ZCY is also a right angle as it creates a pair of supplementary angles with ∠DCB.
To find the value of x, we can use the Pythagorean Theorem. To do this let's recall that the sum of squared legs of the right triangle is equal to its squared hypotenuse. In △ZCY, the legs are x+ 7 and x, and the hypotenuse is 13.
We will use the Quadratic Formula to solve the given quadratic equation.
ax^2+ bx+ c=0 ⇔ x=- b± sqrt(b^2-4 a c)/2 a
Now, we can identify the values of a, b, and c.
x^2+7x-60=0 ⇔ 1x^2+ 7x+( -60)=0
We see that a= 1, b= 7, and c= -60. Let's substitute these values into the Quadratic Formula.
c Using the formula for the area of a square we recalled in the previous parts, we know that if the area of a square is g square meters, the side length of this square is sqrt(g) meters. Let's use this information to find the lengths of sides that extend the side length of ABCD. We will use the fact that AB=2CY.
Now let's look at the given picture. Let x be the side length of a square WXYZ.
We can solve for x using the Pythagorean Theorem, as we know from the previous parts that △ZCY is a right triangle. In this exercise, legs of the triangle are sqrt(g)2+ g and sqrt(g), and the hypotenuse is sqrt(g)2.
