The foot of an extension ladder is 9 ft from a wall. The height that the ladder reaches on the w

The foot of an extension ladder is 9 ft from a wall. The...

Key Concepts

Pythagorean Theorem

A fundamental principle in geometry that relates the three sides of a right triangle, stating that the square of the hypotenuse equals the sum of the squares of the other two sides. This theorem is essential for solving problems involving distances and relationships between lengths in right-angled triangles.

Consecutive Integers

These are numbers that follow each other in order with a difference of one between each pair. In problems where two quantities are described as consecutive integers, one can define one variable and express the other as that variable plus one, forming a basis for setting up algebraic equations.

Quadratic Equation

A type of polynomial equation of the second degree that arises when variables are squared. Quadratic equations are commonly solved by factoring, completing the square, or applying the quadratic formula, and they are instrumental in finding unknown quantities in geometric and algebraic problems.

Transcript

00:01 All right, so the question we're looking at today, we have a foot, the foot of an extension ladder is nine feet away from the wall.

00:07 And the height this ladder reaches on the wall and the length of the ladder are consecutive integers.

00:11 So consecutive integers would be something like 11 and 12 or 8 and 9.

00:16 And we need to figure out how long the ladder is.

00:19 So we're going to draw a sketch from the information we're given.

00:22 We have this is going to be our ladder.

00:25 This is going to be the base.

00:27 And then this is going to be the height it reaches on the wall.

00:31 So we know that the foot of the extension ladder is nine feet away from the wall.

00:36 So this right here is all going to be nine feet.

00:39 Then we have that the height of the ladder reaches on the wall, and the length of the ladder are consecutive integers.

00:45 So if the height of this was x, then the hypotenuse of it, or the length, would have to be x plus 1, because it would have to be greater than the height.

00:57 And this is what the information that we're going to use to create our form.

01:01 And solve for the how long the ladder is.

01:05 So we're going to put this into pythagorean theorem, where our a is going to equal our x plus 1, or no it's not, it's just going to equal x.

01:18 A is going to be our x.

01:20 B is going to be our 9, and c is going to be x plus 1.

01:28 Plugging that all in, we get x squared plus jumping ahead, 9 squared is equal to x plus 1 squared...

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