CHANGING DIMENSIONS Each pair of figures is similar. Find the area and perimeter of each figure.

CHANGING DIMENSIONS Each pair of figures is similar. Find...

Key Concepts

Effects of Dimension Changes

Changing the dimensions (scaling) of a figure impacts its geometric properties differently. While the perimeter, being a linear measure, changes linearly with the scale factor, the area, being a two-dimensional measure, changes quadratically. This means that even small changes in dimensions can result in significant changes in area compared to the corresponding change in perimeter.

Area

The area measures the amount of space inside a figure and depends on the square of its dimensions. When a figure's dimensions are increased or decreased by a scale factor, the area changes by the square of that factor, reflecting a quadratic relationship with the dimensions.

Similar Figures

Similar figures are objects that share the same shape but may differ in size. Their corresponding angles are equal, and their corresponding sides are proportional. This property is fundamental when understanding how scaling affects various measurements like area and perimeter.

Scale Factor

The scale factor is the ratio by which all linear dimensions of a figure are multiplied to produce a similar figure of a different size. It directly affects the length of sides and the perimeter, and it influences the area by the square of the scale factor due to the two-dimensional nature of area.

Perimeter

The perimeter of a shape is the total length around its boundary and depends linearly on the dimensions of the figure. When the dimensions of a figure are scaled by a factor, the perimeter is also scaled by the same factor, demonstrating a direct proportional relationship.

Transcript

00:01 All right.

00:03 For this problem, we want to describe the relationship between similar triangles and then the relationship between similar triangles perimeter as well as their areas.

00:16 And if there is a relationship or not.

00:19 So you're given an equilateral triangle.

00:22 The first triangle has sides of four.

00:25 And the second triangle has another equilateral triangle all has sides of five.

00:30 So the scale of fact.

00:32 So the scale of fact.

00:32 In order to get from a four to a length of five means that they multiplied every side by 1 .25.

00:39 And so if they multiplied every side by 1 .25 to get a similar triangle, which is also equilateral, and there is a relationship between them, it's just increased by 1 and a quarter of the amount.

00:53 So the scale factor here is 1 .25 if i go in that direction.

00:57 If you were comparing a 4 to 5 ratio, then it is 0 .8.

01:03 All right, so i'm going to continue to go in the order of i'm looking at the larger side or the larger triangle first and then the smaller side length triangle second.

01:14 I want to find the perimeter of these two.

01:16 So in order to find perimeter, you just add up all of the side lengths together.

01:21 If i add up all the side lengths of the second triangle with side lengths of five, i'm going to get a perimeter of 15.

01:28 And if i add together all the side lengths up with the triangle with side lengths of four, you are going to get a perimeter of 12.

01:39 I just want to check, well, what's like the ratio between those? and the ratio is kind of like a scale factor here.

01:44 And what would that scale factor be? is their relationship between them? and if you do 15 divided by 12, you'll notice that the scale factor between the perimeters is also 1 .2.

01:58 Five the exact same scale factor between the side lengths.

02:04 All right, on top of that, i want to now compare areas.

02:08 So i notice that there is a relationship between if i multiply the sidelines by a scale factor, then that scale of factor is going to apply to the perimeters.

02:19 Now, what about the areas for these? so i'm going to write the area values in green.

02:27 And just to label red was the perimeter.

02:32 All right, area is going to be in green.

02:45 So if i was to find the area of this triangle, you are going to have to do a little bit of work in finding this.

03:00 So for the sidelines where these are all five, you have to find the height.

03:09 In order to find the height, you're going to go through and use the pythagorean theorem to find that missing height value.

03:14 Now the height cuts the base of the triangle exactly.

03:17 And a half.

03:18 So instead of this being an entire length of five, it's really now just a length of two and a half.

03:24 And you need to find this height.

03:25 You know what the hypotenuse is.

03:27 It's five because it was an equilateral triangle...

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