Tile Design. The Old World tile company wants to make a new rectangular tile that has a perimete

Tile Design. The Old World tile company wants to make a new...

Key Concepts

Systems of Equations

When a problem provides multiple conditions, such as a specified perimeter and diagonal length, determining the dimensions of the rectangle involves setting up and solving a system of equations. Each equation represents a different geometric property, and solving them simultaneously requires algebraic manipulation or substitution. This process is an essential technique in algebra that applies to a wide range of scenario-based problems in geometry.

Rectangle Properties

Rectangles are quadrilaterals with opposite sides equal and all interior angles equal to 90 degrees. This property allows for the use of specific formulas related to their geometry, such as the formulas for perimeter and the diagonal. Understanding the basic structure of a rectangle is essential in approaching problems related to dimensions and shape characteristics.

Perimeter of a Rectangle

The perimeter of a rectangle is calculated by summing the lengths of all four sides, which can be efficiently expressed as twice the sum of its length and width. This formula, P = 2(l + w), is widely used in geometry for determining the boundary lengths of rectangular shapes, and it forms a foundational concept in problems where the total boundary measure is known.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle used in geometry, stating that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In the context of a rectangle, the diagonal forms the hypotenuse of a right triangle created by the length and width, allowing the relationship l² + w² = d² to be established. This relationship is critical when the diagonal length is involved in the calculation of a rectangle's dimensions.

Transcript

00:02 Okay, so we are provided a scenario where we have a tile that's perimeter is 6 inches and the diagonal is root 5 inches and we need to find the dimensions of this tile.

00:13 So the best way to start is first to draw a picture.

00:19 So it's most likely going to be rectangular.

00:22 We know the diagonal is root 5.

00:26 And we're going to have a length x or a width x and the length y.

00:31 So we know the perimeter, our x plus x plus y plus y or 2x plus 2y, is equal to 6 inches.

00:40 And then by the pythagorean theorem, we know, since this is a right angle, we know the x squared plus y squared is going to equal our diagonal squared.

00:49 So x squared plus y squared is going to equal root 5 squared or just 5.

00:56 Now we can go ahead and solve this through substitution by solving for either x or y in our first equation.

01:02 So i'm going to subtract by 2y on the top.

01:06 So i'll have 2x equals 6 minus 2y and divide by 2.

01:12 So we're going to have x equals 3 minus.

01:20 And then we can plug that into our second equation.

01:24 So we'll have 3 minus y squared plus y squared equals 5.

01:31 Let's expand out those parentheses by doing 3 minus y times 3 minus y plus y equals 5 and let's foil that out.

01:43 So 3 times 3 is 9, minus 3 times y...