The diagonal braces in a lookout tower are 15 ft long and span a horizontal distance of 12 ft. H

The diagonal braces in a lookout tower are 15 ft long and...

Key Concepts

Right Triangle

A right triangle is a triangle that has one 90-degree angle. It serves as a fundamental shape in geometry where the relationship between its sides is governed by specific rules such as the Pythagorean theorem.

Pythagorean Theorem

The Pythagorean theorem is a key concept in Euclidean geometry which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem is essential for determining an unknown side when the other two sides are known.

Decomposition into Horizontal and Vertical Components

Decomposing a diagonal or hypotenuse into horizontal and vertical components involves understanding how a line can be split along orthogonal axes. This concept is vital in solving problems where the separate contributions in each direction must be determined, often using the Pythagorean theorem or trigonometric functions.

Transcript

00:05 So in this problem, we were given braces on a tower, and the pieces of the braces we were given was a diagonal peace such as this that is 15 feet and then a horizontal piece that was 12 feet.

00:26 This is given information, and we know that it's a right triangle.

00:31 So what we're trying to find is the length, which is in green.

00:36 We're trying to find x using our pythagorean stare.

00:40 Um, pythagorean serum states that the squares of the two legs of a right triangle, the axe and the 12 when squared and then added together will equal the square of the high pot news, which in this case is 15.

00:56 So there's my right angle.

00:57 The two legs are adjacent to the right angle.

01:01 The high pot news is always opposite the right angle.

01:04 The high pot news always is.

01:07 See the two legs accent 12 can be a or b.

01:13 It doesn't matter the legs, because addition is communicative.

01:17 The legs ca nbi either one in either order.

01:21 But what must stay true is the 15 is the high pot news.

01:25 Who is seeking because reading left to right.

01:27 I see the experts.

01:29 I'm gonna substitute my axe for a i'm then gonna substitute my 12 or be and i'm gonna substitute my 15 her seat.

01:42 I want to sell for acts, but i need to start to clean things up a little bit before i start the solving process.

01:48 What i can clean up is squaring the 12 and squaring the 15.

01:52 So let me go ahead and do that.

01:54 I cannot square the action, so he will stay x squared plus 12 squared is 144 15 squared is 225.

02:06 We are in a polynomial factoring chapter.

02:11 There were other ways you can solve for x in this case, but since we are learning to factor, what i'm going to do is bring everything from the right side to the left, set it equal to zero, then factor use my zero product property and find my solutions.

02:30 So in that case, by setting the right side equal to zero, i'm gonna take away the to 25 from both sides when i due to the right, i must do to the left.

02:43 And i'm gonna go ahead and clean that up and i will have ax squared minus 81.

02:52 That's an 81.

02:52 I apologize.

02:53 That kind of list messy equals zero.

02:56 It's an 81.

02:59 So i'm looking at me.

03:01 Let me erase that 81 a little bit because he kind of looks a little bit weird.

03:06 81.

03:07 There we go x squared minus 81 equals zero.

03:10 We've learned different factoring strategies.

03:13 Want factoring strategy we have learned at this point is called the difference of two perfect squares.

03:19 What is required in that is that there is a difference.

03:22 There's a minus sign in the left side, and the either term on either side is a perfect square.

03:31 Can i take the square root of x squared? yes, i can't.

03:34 And when i take the square root of x squared, i get x.

03:39 So both factors air going to start with the square root of x, the difference of two perfect square states that one factor will be adding the other factor subtracting again...

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