The length of one leg of a right triangle is three feet more than the other leg. If the hypotenu

step 1 So for this question, we're given a right triangle. It tells us that the length of one of the le

The length of one leg of a right triangle is three feet more...

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Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in Euclidean geometry that establishes a relationship among the three sides of a right triangle. It states that the sum of the squares of the two legs is equal to the square of the hypotenuse, which serves as the basis for solving problems involving right triangles.

Algebraic Modeling

Algebraic modeling involves translating real-world or geometric situations into algebraic expressions or equations. In the context of right triangles, this often means representing the lengths of sides with variables and forming equations that capture the relationships described in the problem.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree and typically take the form ax² + bx + c = 0. They are integral to solving problems derived from the Pythagorean Theorem, where substituting variables for the triangle's side lengths results in quadratic equations that must be solved to find unknown measurements.

Transcript

00:01 So for this question, we're given a right triangle.

00:04 It tells us that the length of one of the legs is three feet more than the other leg.

00:11 So i'm going to call the shorter leg.

00:13 I'm just going to call that x.

00:15 And then for the longer leg, that's going to be three feet more than the x leg.

00:21 So that's going to be x plus three.

00:23 And then they tell us that the hypotenuse is 15 feet.

00:28 So now that we have all of our sides labeled, we can use the pythagorean 3.

00:33 Theorem.

00:34 We have a, b, and c.

00:37 So that's a squared plus b squared equals c squared.

00:43 So in this case, we have x squared plus x plus 3 squared equals 15 squared.

00:55 So we can go ahead and square all of this out.

00:58 So we have x squared plus x plus 3 squared is going to be x squared plus 6x plus 9.

01:08 And then 15 squared is 225.

01:14 So now i'm going to go ahead and move the 225.

01:20 I'm going to subtract that over to the other side and also combine the like terms that are on the left.

01:29 We can do that in the same step.

01:31 So when we do that, we get 2x squared plus 6x, and then 9 minus 225 is negative 216, and that equals 2, equals 0.

01:47 So now that we have the equation set up like this, we can go ahead and factor...

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