Dimensions of a Rectangle A circular piece of sheet metal has a diameter of 20 in. The edges are

Dimensions of a Rectangle A circular piece of sheet metal...

Key Concepts

Inscribed Figures

This concept involves understanding how a rectangle can be positioned within a circle so that all its vertices lie on the circumference. In such cases, the rectangle's diagonal is equal to the circle’s diameter, which creates a direct relationship between the dimensions of the rectangle and the dimensions of the circle.

Pythagorean Theorem

The Pythagorean Theorem is crucial in relating the sides of the rectangle to its diagonal, especially when the rectangle is inscribed within a circle. Since the diagonal of a rectangle with length and width is given by the square root of the sum of the squares of these two dimensions, this theorem helps in formulating the equation connecting the rectangle’s sides and the circle’s diameter.

Area of a Rectangle

The area of a rectangle is computed by multiplying its length by its width. In problems where the area is given along with other geometric constraints, this concept is used to set up one of the equations in the system, which then helps in determining the dimensions of the rectangle.

System of Equations

In this type of problem, multiple conditions such as the area of the rectangle and the relationship between the rectangle’s diagonal and the circle's diameter lead to a system of equations. Solving these simultaneous equations is key to determining the unknown dimensions of the rectangle. This often involves substitution or elimination methods to arrive at the solution.

Transcript

00:01 So on this problem, we're given some information about a rectangle that is inscribed in a circle.

00:06 And our job is to create and solve a system of equations that allows us to find the dimensions of this rectangle.

00:15 So we're given some information here.

00:17 First, we're told that the diameter of the circle is 20 inches.

00:22 Now, we can place a diameter anywhere on the circle as long as it goes straight across from one edge of the circle to the other.

00:30 So what's really neat here is to place it on the diagonal of the rectangle.

00:35 So then the diameter of the circle can now actually be a role in this rectangle.

00:42 So using that information, we can then use the pythagorean theorem, as we have a right triangle going on here.

00:49 To write x squared plus y squared equals 20 squared, which is 400, where x and y are the sides of the rectangle.

01:03 Okay, so that's going to help us.

01:04 And then the other piece of information that we have is that the area of this rectangle is 160 square inches.

01:12 So that means that x times y is equal to 160.

01:19 You can use x or y.

01:21 You can use l or w, whatever you like.

01:24 All right.

01:25 So now we have right here a system of two equations and two variables.

01:29 So we can start taking some steps to solve the system, find x and y.

01:34 So it looks like substitution is going to be our best bet here.

01:38 So let's solve this bottom equation for either x or y, and then substitute it in in the first equation.

01:45 So if we solve for y here, you can do it either way.

01:48 But if we solve for y, we get y equals 160 over x.

01:54 And now we can substitute this y in to the top equation.

01:59 It's going to be helpful to have a calculator here as we work with some of these larger numbers.

02:03 But let's get into that.

02:05 So when we make that substitution, we have x squared plus 160 over x squared equals 400.

02:19 And then squaring this term, 160 squared is 25 ,600.

02:28 That's over x squared.

02:32 That equals 400.

02:35 Now, in order to move forward with solving this, we really need to get this x squared out of the denominator.

02:41 So to do that, let's multiply both sides by x squared.

02:46 So that means that our first term becomes x to the fourth...

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