Find the lengths of the sides of the following right triangles. (Hint: Use the Pythagorean theor

Find the lengths of the sides of the following right...

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that establishes a relationship between the three sides of a right triangle. It states that in any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This theorem is essential for solving problems involving the computation of an unknown side length in right triangles.

Right Triangle

A right triangle is a type of triangle that includes one angle of exactly 90 degrees. This property allows the application of specialized rules, like the Pythagorean Theorem, to determine lengths and relationships between the sides. Understanding the properties of right triangles is crucial in various mathematical and real-world applications.

Legs and Hypotenuse

In a right triangle, the two sides that form the right angle are known as the legs, while the side opposite the right angle is called the hypotenuse. The legs and the hypotenuse have a unique interrelationship defined by the Pythagorean Theorem, making it possible to calculate the length of any missing side provided the lengths of the other two are known.

Transcript

00:01 We're given a right triangle with sides 2x, x plus 5, and 3x plus 1, and so we're asked to find the actual length of the side.

00:08 In order to do that, we need to find a value of x.

00:11 So we're going to use pythagrin theorem.

00:13 So a squared plus b squared is equal to c squared.

00:16 C is always across the right angle.

00:19 So this is c.

00:20 A and b.

00:22 It doesn't matter which is a or b.

00:24 So i'm going to have 2x squared plus x plus 5 squared is equal.

00:31 To 3x plus 1 squared.

00:35 So 2x squared is 4x squared.

00:38 X plus 5 squared is you use the square of a binomial, so you take the first term in squared, so that's x squared.

00:47 To get the second term, you multiply x and 5 and double it, so you get 10x.

00:52 To get the last term, you take a 5 and you square that, and that's a plus 25.

00:58 Do the same thing with the other side.

01:00 3x squared becomes 9x squared, we multiply the first term and the last term, so that's 3x.

01:08 Then you double it, so that's 6x.

01:11 Then we take the last term and we square it, and that becomes 1.

01:17 So we're going to combine like terms.

01:20 I have 5x squared plus 10x plus 25 is equal to 9x squared plus 6x plus 1.

01:29 I'm going to move everything from the left side to the right side so i can keep the x squared positive.

01:35 So i'm going to subtract x squared.

01:38 I'm going to subtract 10x and make sure you line it up with the common terms and subtract 25.

01:46 So this all goes to zero and you're left with 4x squared minus 4x minus 24.

01:55 Before i factor that, i notice all of these are divisible by four, so i'm going to divide the entire thing by four.

02:02 So i get 0 is equal to x squared minus x minus 6.

02:07 Makes it easier to factor.

02:10 So i know i'm going to have two parentheses equal to zero...

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