Consider the trapezoid below. If its area is a^2+2 a b+b^2,...

Key Concepts

Perfect Square Trinomial

This involves recognizing that an expression like a^2 + 2ab + b^2 is a perfect square trinomial that factors into (a + b)^2. Recognizing this pattern simplifies the problem by allowing the substitution of the factorized form into other equations, streamlining the algebraic manipulation.

Algebraic Manipulation

This encompasses the process of equating two expressions for the same quantity and solving for an unknown variable. In the context of this problem, setting the given area equal to the trapezoid area formula allows for canceling common factors and solving for the height, exemplifying how algebra can be used to manipulate and solve equations.

Trapezoid Area Formula

This concept involves understanding the formula for the area of a trapezoid, which is given by A = 1/2 * h * (base1 + base2). The formula relates the height and the sum of the lengths of the two parallel sides (bases) to the area, making it essential for problems involving trapezoids.

Transcript

00:09 I'm sorry.

00:11 I didn't mean delay there, but i forgot to look to see what we're solving for.

00:14 So we have this problem here.

00:17 A, capital a, equals one -half little a times h plus one -half little b times little h, and we're going to solve for h, which means instead of having a equals, in the end, i want to have h equals.

00:31 That's what solving for h means.

00:34 Well, first thing i want to talk about is the fact that this capital a is different than this little a, because in math, the notation, and how you write things matters.

00:43 So i really don't know what a is.

00:45 I do not know what number it is, but i know that this capital a is a different number than this lowercase a, simply because they're different letters.

00:57 Now, i'm not 100 % sure.

00:58 I mean, they could be the same, but if they're the same number, why not use the same letter? i know h stands for the height.

01:08 I think this problem is with a trapezoid or something.

01:10 I'm not 100%.

01:11 But anyway, we're solving a literal equation and we want to get it solved for h.

01:16 So i'm looking at this and i'm thinking, well, let's try and make it where i only have 1h in the problem.

01:27 So i want to talk about something.

01:28 Let's say you had 5h plus 7h.

01:36 Now, because they both have an h, you can add this part, and that would give you 12h.

01:46 If you had 6v plus 10v, because they both have the variable v, you could add this part and get 16v.

02:00 Another way of writing that is saying, well, 5h plus 7h would be 5 plus 7h, h, would be 5 plus 7h, h, because aren't those two equal? this is 12h and that's 12h.

02:15 6v plus 10v could be 6 plus 10v, because that's 16v, and you know, you did the arithmetic, that's 16b.

02:25 I'm going to use that concept here.

02:29 Because they both have an h, 1 half a plus 1 half b could be written as 1 half a plus 1 half a plus 1 half b, h.

02:45 Again, i'm doing very, very similar to what i did down here.

02:52 Excuse me a second.

02:58 Again, very, very similar.

03:01 While i go, i said if you had 6h plus 3h, you could say that's 9h, or you could say it's 6 plus 3h, which truly does equal 9h.

03:14 So i had one half a h plus one half bh.

03:20 And i said, well, since they both have an h, i can say it's one half a plus one half b h.

03:28 Now at this point, what i'm going to do is i'm still trying to solve it for h.

03:34 So i'm looking at this as this quantity times h.

03:40 This quantity times h...

Please give Ace some feedback