Problem

Twyla has a 26 -ft ladder leaning against her hou…

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Problem 29 Medium Difficulty

The diagonal braces in a lookout tower are 15 ft long and span a horizontal distance of 12 ft. How high does each brace reach vertically?
CAN'T COPY THE IMAGE

Answer

This is a right triangle problem. Since we can draw a picture, we draw a picture. Make a little tower if you want. The diagonal of the rectangle is 15 feet and the horizontal is 12. This makes a right triangle with one leg of 12 and a hypotenuse of 15. The Pythagorean Theorem says that, for any right triangle, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. So, we have 12^2+b^2=c^2. Rearrange and solve to get b=sqrt(15^2-12^2)=9. This is also one of the "standard ratios" of a right triangle, where one leg is a multiple of 3 (9), the other is a multiple of 4 (12), and the hypotenuse is a multiple of 5 (15).

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Algebra

Elementary and Intermediate Algebra

Chapter 6

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Section 7

Applications of Polynomial Equations

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Video Transcript

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. This is given information, and we know that it's a right triangle. So what we're trying to find is the length, which is in green. We're trying to find X using our Pythagorean stare. 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. So there's my right angle. The two legs are adjacent to the right angle. The high pot news is always opposite the right angle. The high pot news always is. See the two legs accent 12 can be a or B. It doesn't matter the legs, because addition is communicative. The legs ca NBI either one in either order. But what must stay true is the 15 is the high pot news. Who is seeking because reading left to right. I see the experts. 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. I want to sell for acts, but I need to start to clean things up a little bit before I start the solving process. What I can clean up is squaring the 12 and squaring the 15. So let me go ahead and do that. I cannot square the action, so he will stay X squared plus 12 squared is 144 15 squared is 225. We are in a polynomial factoring chapter. 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. 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. And I'm gonna go ahead and clean that up and I will have ax squared minus 81. That's an 81. I apologize. That kind of list messy equals zero. It's an 81. So I'm looking at me. Let me erase that 81 a little bit because he kind of looks a little bit weird. 81. There we go X squared minus 81 equals zero. We've learned different factoring strategies. Want factoring strategy we have learned at this point is called the difference of two perfect squares. What is required in that is that there is a difference. There's a minus sign in the left side, and the either term on either side is a perfect square. Can I take the square root of X squared? Yes, I can't. And when I take the square root of X squared, I get X. 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. It doesn't matter because multiplication is communicative. If the first factor is X minus in the second factor is X plus. That's okay. Take the square root of 81. It's nine. So one of these factors gets a nine and sodas the other. The Onley difference in the two factors is the plus in the minus sign. That is a true difference of two perfect squares. Why did it perfectly factor that way? If you foil this, if you distribute and multiply these factors through, you will achieve expert minus 81. You're to middle terms nine x and minus nine x The linear X will cancel itself out through foiling. So we have two factors. The zero product property states that if you are multiplying Ah, whole bunch of factors together, all it takes is one factor to become a zero. And the rest of the left side is all zero and it's true. So I have one factor. My first factors X plus nine and my second factor is X minus nine. I have two factors. Either factor can be zero, and this will be true. So what I'm gonna do is create the two scenarios. One scenario says if this first factor is zero, then so is my entire left side. And it is all a true equation. So experts nine if that zero this is gonna be a perfectly beautiful equation. I will subtract the nine from both sides and I get my first solution. Off acts equals negative nine. I will get to solutions. This is a quadratic. It takes the shape of a parabola. The solutions are dictating where the X intercept is gonna be. And there will be two X intercepts, typically with the parabola. So now my second factor, if that happens to be equal to zero, so is the entire left site. So I'm gonna solve for X in this case at nine to both sides, and I get X equals nine two solutions Negative nine and positive night. Now what we need to do is return to the word problem and realize, huh? We're looking at lengths of a triangle. There is only one valid solution that answers the question. Now, these air both solutions. But only one answers the question. You cannot have negative nine feet in length, so we will throw out this solution, invalidate it for this particular problem. And this we have a solution of nine feet acts is nine feet. So if we go back up to here, we can fill in nine feet. If we turn around and plug nine squared plus 12 squared equals 15 squared, it will totally be a true equation. Beautiful

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