Archaeologists have discovered that the 18 th-century garden of the Charles Carroll House in Ann

Archaeologists have discovered that the 18 th-century garden...

Key Concepts

Quadratic Equations

When using the Pythagorean theorem in combination with additional linear relationships between a triangle's sides, the result is often a quadratic equation. Solving this equation requires understanding how to manipulate and find the roots of quadratic expressions, ensuring that only solutions that make sense in the geometric context (e.g., positive side lengths) are considered.

Algebraic Formulation in Geometry

Many geometric problems require translating physical or spatial relationships into algebraic equations. By assigning variables to unknown measurements and using given conditions, such as differences in side lengths, one can set up equations that capture the problem's essence and solve for the needed values.

Right Triangle Fundamentals

A right triangle is a triangle with one angle measuring 90 degrees. The two sides that form this right angle are known as the legs, and the side opposite the right angle is the hypotenuse. This concept is foundational in geometry as it establishes the relationship between the sides using the right angle.

Pythagorean Theorem

The Pythagorean theorem is a key principle associated with right triangles. It states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. This theorem is essential for determining unknown side lengths in right triangles through algebraic relationships.

Transcript

00:01 In this particular problem, we are looking at a right triangle, which means pythagorean's theorem is going to come into play.

00:08 Now, what is given on this right triangle is a 400 -foot leg right here.

00:20 In this problem, they make reference to the fact that the hypotenuse is 200 more than the other leg.

00:26 We can't identify the hypotenuse until we define the other leg.

00:31 So what we are going to do is create a name for the other leg.

00:35 More than likely, you will use x.

00:38 Variable, you can choose any variable to represent that leg, but more often than not, we use an x.

00:43 So that's what i'll use.

00:45 And once i identify the leg, then i can make use of the information that the hypotenuse is 200 more than the other leg, which is x.

00:59 Here is our right angle.

01:01 So x plus 200 is the hypotenuse.

01:06 Pythagorean's theorem states that the sum of the square of two legs of a right triangle will always equal the square of the hypotenuse.

01:17 You have no choice but that c will be substituted with x plus 200.

01:23 The a and b can be interchanged because addition is commutative between the x and the 400.

01:29 It's your choice.

01:31 Because i read left to right, i'm going to substitute the x in for my a because that's the first leg that i see left to right.

01:39 But again, it makes no difference.

01:41 And i'm going to substitute the 400 in where the b is.

01:45 If you do 400 squared plus x squared, you're totally 100 % okay.

01:50 What we have to keep in mind is how careful we need to be with this hypotenuse.

01:54 What we see here is a binomial.

01:58 Right here, i'm going to wrap this in parentheses.

02:01 A binomial squared is not, this particular binomial will not be x squared plus 40 ,000.

02:09 No, this must be foiled.

02:12 X plus 20 in parentheses times x plus 20 in parentheses and foiled.

02:18 So i'm going to put that in parentheses to ensure that he gets foiled.

02:29 And foiling means this is x plus 20 times 200 plus x plus 200 times.

02:35 X plus 200.

02:37 I'm going to rewrite the rights.

02:39 I'm going to scoot over a little bit to make room on my page plus 400 squared is, oops, my bad there, it is, let me erase that, 160 ,000.

02:56 160 ,000.

03:00 X plus 200, the quantity squared.

03:03 My first term is x times x, which is x squared.

03:06 The second terms will be the x times 200, which is 200x.

03:15 The third term is 200 times x, which is 200x.

03:19 I combine those two terms together, and i will get a 400x.

03:24 So 400x is a result of 200x plus 200x.

03:30 Then i multiply my last terms, 200 times 200, and get 40 ,000.

03:38 So what we're staring at is a polynomial.

03:42 Multiple terms on both sides of the equal sign.

03:46 A quadratic because x is squared.

03:50 So the goal is to make one side completely a zero and bring all like terms over to the other side and combined like terms so that you have a polynomial equal to zero.

04:03 And then we begin the factoring process.

04:06 So it doesn't matter which side you...

Please give Ace some feedback