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Suppose two triangles are similar to each other and the first triangle has a side of length 8 inches and its area is 24 square inches. If the corresponding side of the other triangle is $12,$ what is its area?

54 sq in

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 9

Elements of Geometry

Derivatives

Missouri State University

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

04:19

GEOMETRY The hypotenuse of…

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Each side of an equilatera…

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Find all right triangles f…

02:57

The congruent sides of an …

00:38

Area of a triangle: Using …

00:36

Solve each problem. See Se…

for this problem, we have to similar triangles. These problems are really good. To start with a diagram, you do not have to be a great artist to draw two triangles. Speaking able to see them will help us make sure that we have everything written down properly. So we're not missing anything when we compare our triangles. Okay. What do we know about these triangles? Let's put some numbers on here. We're told the first triangle has a side of length eight. Well, since I know we're gonna be talking about area, let's make the base B eight because that's part of finding the area of a triangle. Remember, Area of a triangle is one half the base times the height. Okay, so we're going to call the base here eight. We also know that it's area is 24 square inches. Okay, Now we go to our corresponding triangle or similar triangle. The corresponding side there is 12 inches, and we want to find what it's area is. Okay. So, back to our first triangle, we know that area is one half the base times the height. So in this case, are areas 24 and that's going to be one half the base, which is eight times are height. Now, we don't know our height, so I'm just gonna put an h there. Half of eight is four. Divide both sides by four. That gives us a height of six. Okay, now, when we go from one triangle to the next, every length has is going to either grow or shrink by the same proportion. That means that sides and its height. So if I want to find the height of this second bigger triangle, I can look at how the sides changed. The height will change with the same proportion. So small to big is 8 to 12. Okay, now, what about my heights? Small to big is six to the height of this big triangle. So 8/12. I can simplify this to to over three. We might as well keep our numbers as small as possible. Cross multiply. I get to H equals 18 or H equals nine. Okay, so now I have what I need to find the area of my large triangle. The area equals one half the base times the height half of 12 and six and six times nine is 54. So it's going to be 54 square inches

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