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QuestionFind the length of a side of square ABCD if its area is 89 cm². What is xundefined\widehat{x}?

Studdy Solution

STEP 1

Assumptions1. The area of the square ABC is89 cm². We need to find the measure of the angle x3. The angle x is the angle between a diagonal of the square and one of its sides

STEP 2

First, we need to find the length of a side of the square. We can do this by taking the square root of the area.
Side length=AreaSide\, length = \sqrt{Area}

STEP 3

Now, plug in the given area to calculate the side length.
Side length=89 cm2Side\, length = \sqrt{89\, cm^{2}}

STEP 4

Calculate the side length.
Side length=89 cm2=9.43 cmSide\, length = \sqrt{89\, cm^{2}} =9.43\, cm

STEP 5

Now that we have the side length, we can find the length of the diagonal of the square. The diagonal of a square is given by the formula Diagonal=Side length×2Diagonal = Side\, length \times \sqrt{2}.

STEP 6

Plug in the side length into the formula to calculate the diagonal.
Diagonal=9.43 cm×2Diagonal =9.43\, cm \times \sqrt{2}

STEP 7

Calculate the diagonal length.
Diagonal=9.43 cm×2=13.34 cmDiagonal =9.43\, cm \times \sqrt{2} =13.34\, cm

STEP 8

Now that we have the side length and the diagonal length, we can find the measure of the angle x. The angle x is the angle between the diagonal and the side of the square, which can be calculated using the formula for the cosine of an angle in a right triangle cos⁡(x)=Adjacent sideHypotenuse\cos(x) = \frac{Adjacent\, side}{Hypotenuse}.

STEP 9

In this case, the adjacent side is the side of the square and the hypotenuse is the diagonal. Plug in these values into the formula to calculate the cosine of the angle x.
cos⁡(x)=9.43 cm13.34 cm\cos(x) = \frac{9.43\, cm}{13.34\, cm}

STEP 10

Calculate the cosine of the angle x.
cos⁡(x)=9.43 cm13.34 cm=0.707\cos(x) = \frac{9.43\, cm}{13.34\, cm} =0.707

STEP 11

Now that we have the cosine of the angle x, we can find the measure of the angle x in degrees by using the inverse cosine function.
x=cos⁡−(0.707)x = \cos^{-}(0.707)

STEP 12

Calculate the measure of the angle x.
x=cos⁡−(0.707)=45 degreesx = \cos^{-}(0.707) =45\, degreesThe measure of the angle x is45 degrees.

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