One leg of a right triangle is 7 cm less than the length of the other leg. The length of the hyp

One leg of a right triangle is 7 cm less than the length of...

Key Concepts

Right Triangle

A right triangle is a triangle in which one angle is exactly 90 degrees. The two sides forming the right angle are called legs, and the side opposite the right angle is called the hypotenuse. This concept is fundamental when exploring relationships between the sides of the triangle, especially through the Pythagorean theorem.

Pythagorean Theorem

The Pythagorean theorem is a central principle in geometry that defines the relationship among the three sides of a right triangle. It states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides, forming the equation a² + b² = c². This theorem is widely used to solve problems involving right triangles, such as finding missing side lengths.

Algebraic Substitution

Algebraic substitution is a method used in solving equations where one variable is replaced by an equivalent expression in terms of another variable. This technique simplifies the solving process by reducing the number of variables and is especially useful in problems where relationships between quantities are expressed algebraically.

Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically taking the form ax² + bx + c = 0. In the context of right triangle problems involving the Pythagorean theorem, the equation often simplifies into a quadratic form that can be solved by factoring, completing the square, or using the quadratic formula. Understanding quadratic equations is essential for finding valid solutions in these scenarios.

Transcript

00:01 Hello, so we have this situation here and is dealing with a right triangle.

00:06 So let's draw a right triangle.

00:10 And we know that the hippanos is 13 centimeters.

00:14 And then one of the legs is 7 meters less than the lump of the other leg.

00:22 So we're going to have x minus 7 and then x here.

00:26 Now we're going to plug in our values in the pythagorean theorem, which is a square plus b squared, equal c square.

00:36 So our a square is going to be x minus seven squared, and then we have b square, which is x squared, and then c square, which is 13 squared.

00:48 We're going to expand this x minus seven squared, so that will be x squared minus 14x plus one.

00:57 49 and bring down the x squared and then 13 squared is 169 so then now we're going to combine these two x squares together and get 2x squared minus 14x plus 49 which equals to 169 our next step is bringing this 169 to the left hand side so we also attract 169 on both sides and we'll get 2x squared minus 14x minus this one no it will be let's see yes it will be 120 sorry next we're going to divide these terms by two so let's do that both sides and we'll get x squared minus 7x minus 120 which equals to 0 and then we're going to oops this is going to be 60.

02:29 Now we're going to split this negative 7x and we will have x squared minus 12x plus 5x minus 6x minus 60 equals 0.

02:55 And then we will have, we're going to split these up.

03:01 We're going to factor out the x here.

03:04 And we have x minus 12.

03:07 And then here we're going to factor out the 5...

Please give Ace some feedback