The diagonal of a rectangle measures 10 inches. If the width is 2 inches less than the length, t

The diagonal of a rectangle measures 10 inches. If the width...

Key Concepts

Rectangle Geometry

Rectangle geometry is a branch of geometry that focuses on the properties and relationships specific to rectangles, such as the connection between their side lengths, diagonals, and area. Understanding rectangles provides a foundation for exploring how linear dimensions interact and how geometric shapes can be analyzed using algebraic methods.

Pythagorean Theorem

The Pythagorean theorem is a central concept in Euclidean geometry that relates the lengths of the sides of a right triangle. In the context of rectangles, the diagonal forms the hypotenuse of a right triangle whose legs are the rectangle's length and width, allowing the use of the theorem to determine unknown dimensions.

Quadratic Equations

Quadratic equations are algebraic expressions where the highest exponent of the variable is two. They often arise in geometric problems after applying the Pythagorean theorem, leading to equations that must be solved to find the length and width of a rectangle.

Area Calculation

Area calculation involves computing the measure of the space inside a shape. For a rectangle, the area is found by multiplying its length by its width. Mastery of area calculations is fundamental in geometry, as it connects algebraic expressions with physical dimensions.

Transcript

00:01 Our next application for solving quadratic equations, we're given a square, i'm sorry, a rectangle, pardon me, a rectangle, and we're given the length of the diagonal, so the distance from one corner to the other, and that is 10 inches.

00:17 We're also given the length, which is l, and that the width is equal to two less than the length, so l minus two.

00:28 And all this is in inches, obviously.

00:31 How to find, we want to find the area of this rectangle.

00:35 So the area of the rectangle is equal to the length times the width.

00:43 But this doesn't help us yet because we don't have any numbers to work with.

00:47 So in order to figure out what the length and the width are equal to, we have to remember that this is a right triangle that we're looking at.

00:55 When we split this rectangle and a half is a right triangle.

00:57 So we can use the pythagorean theorem to help us solve for what the lengths are.

01:03 Pythagorean theorem says that the two legs of a triangle, of a right triangle, a squared plus b squared, is equal to the hypotenuse squared.

01:13 So in this case, our two legs are our length and our width.

01:16 So say l squared plus w squared is equal to the hypotenuse squared.

01:23 And the hypotenuse is 10, so 10 squared.

01:26 We also know that the width is equal to the length minus 2.

01:32 So l minus 2 squared and 10 squared is 100.

01:39 Let's simplify.

01:40 So l squared plus l minus 2 all squared.

01:45 So that gives us l squared minus 4l plus 4 and that equals 100 again.

01:54 And now let's just combine some like terms...

Please give Ace some feedback