Perimeter & area (video) | Area | Khan Academy
As can be seen in Similar Triangles - ratios of parts, the perimeter, sides, altitudes and medians are all in the same ratio. Therefore, the area ratio will be the. Learn how to find the perimeter and area of similar polygons by using the Notice that the only difference between the perimeter of the first triangle and the. What is the relationship between perimeter, circumference, area, surface area, and/or volume of similar Recall the relationship between scale factor and perimeter ratios of similar figures. . Use words, symbols, or both in your explanation. c.
Each group should generate its own rule, and then test the rule on the three rectangles and three triangles in their logs. Does the rule they generated work for all six figures? If not, ask students to rework the rule and try again. Emphasize that the rule will only be useful if it works for all figures, all the time!
How Do You Use Similar Figures to Find the Area of a Polygon? | Virtual Nerd
In this step, students are just putting the last statement completed on the triangle log sheet into their own words. What will the teacher do to bring the lesson to a close? How will the students make sense of the investigation?
Teacher returns to the similar rectangles question from the "hook. Tell them to write their explanation in terms of the scale factor of the rectangles. The teacher should be sure that all students realize that if two 2D figures have a scale factor ofthen their perimeters will differ by a factor ofand their areas will differ by a factor of.
Summative Assessment Teacher will draw a new set of similar rectangles and triangles on the board. Ask students to identify the scale factors that relate the similar rectangles and the scale factor that relates the similar triangles. Next, ask for volunteers to predict how the area and perimeter of figures will change in relation to their scale factors; e.
Have them justify that the triangles they have drawn are indeed similar. Then have them identify the scale factor that relates the two triangles. Formative Assessment Examine the four triangles in the figure below. Downloadable attachment Measure all three sides and all three angles of each triangle. Make three observations about the measures you found.
What is different about certain measures in the third triangle? Teacher looks for observations that capture the following ideas: The two triangles in the first row have corresponding side lengths that are the same, and their corresponding angles are the same. The triangle in the second row has sides that are twice as long as the corresponding sides of either triangle in the first row, but corresponding angles are unchanged.
The angles of the triangle in the third row are different from the corresponding angles of triangles in the first two rows. In addition, its proportions are different from the triangles in the first two rows.
Teacher elicits the definitions of similarity and congruence. Feedback to Students Students will receive immediate feedback when answering questions. The teacher will look for specific areas of difficulty such as: Students may not realize that similarities preserve shape, but not size.
Angle measures stay the same, but side lengths change by a constant scale factor. Students may incorrectly apply the scale factor. For example, students will multiply instead of divide with a scale factor. Students will receive further feedback when they submit their independently produced work the optional homework assignment.
Teacher will draw a new set of similar rectangles and triangles on the board. Large diagrams of each step can be provided. Teacher needs to read all instructions and repeat several times. Ask students to determine the rule that relates scale factor to the volumes of 3D solids that are similar.
For example, if one side of a rectangular solid is 2, and the corresponding side of a similar rectangular solid is 3, then the ratio of their volumes is 8: Emphasize that scale factor multiplies any linear dimension of any 2D object.
For example, if a the length and width of a rectangle are doubled in size, then the diagonal measurement a linear dimension also doubles in size. If a circle has radius one-third the radius of a larger circle, then its circumference and diameter both linear dimensions are also one-third the length of the corresponding value of the larger circle.
Relate the results of this lesson to real-life. For example, doubling the length and width of a home causes floor areas to increase by a factor of four. If the smaller home had 1, square feet, the larger home would have 4, square feet. Since homes are often priced by how many square feet they have, the larger home would be roughly four times as expensive. AC and DF are congruent. And angle A is congruent to angle D. That's angle A, that's angle D.
That's what they tell us. So they just gave us one side and one angle. If they gave us another side, if they said that DE is congruent to AB, that'd be pretty cool. If they gave us this angle, if they said angle F is congruent to angle C, that'd be good. AB is congruent to DE. If AB is congruent to DE then we definitely have congruent triangles.
And you know the theorem that you would have to say in your geometry class is, I have a side, an angle, and a side. So you would say by SAS, by side, angle, side, I know that these two triangles are congruent. So AB is congruent to DE. Let's look at the other ones to make sure we didn't miss anything.
AB is congruent to BC. But that doesn't tell us how AB relates to DE. So that's a useless statement. BC is congruent to EF. Well, see, this is another time that I have a slight problem with the way they're going with this. Because if BC were congruent to EF. Let me think about that. Could I draw this triangle in a way where they're still not congruent.
Because I have this angle here constraining it. They told us that.
So it's not like I can draw this line, FE, coming out here. Because if it came out here, then DE would have to come like that. And then this angle couldn't be what they said it was.
So I'm just trying to think, I actually think that would be sufficient. If you're given that this side is congruent to that side. I think you could make a trigonometric argument very easily to show that these two triangles have equal sides.
But anyway, I'm not going to bother with that. Let's look at choice D.
How Do You Use Similar Figures to Find the Area of a Polygon?
BC is congruent to DE. Well, these aren't even corresponding sides. So that's clearly useless. I have a suspicion that this would have also been enough to prove.
But anyway, I don't want to insult anyone in the California Department of Education, but I'm slightly disappointed by some of these questions. Because I feel like they really aren't testing intuition, they're just testing to see whether you know the definitions of some of these geometric terms.
And whether you can spout out, side, angle, side, and angle, side, angle. And things like that. And you're going to forget those about three hours after you take the test. What's useful is if you know something that gives you an intuition about triangles. That's going to be useful for you on the SAT, that's going to be useful for you when you take trigonometry.
And I'll tell you a dirty secret. Your 9th or 10th grade geometry class is the first and the last time that you'll ever see them.
Area and Perimeter of Similar Figures | Scale Factor Worksheets
So I have a slight problem where they want you to memorize these theorems and all of that. And even some of this notation never shows up again in your mathematical careers. Even if you do a PhD in mathematics. The only time you'll probably see it again is if you become a geometry math teacher. Anyway, but it's good. I mean you should know how to do this stuff at minimum just to jump through that loop that society makes us all jump through.
Anyway, all right, problem And just another aside, I think you can even tell from my tone that I enjoy the SAT problems a lot more. Because in some ways, in fact, in every way, the SAT problems really test your understanding of geometry, but never do they mention the words similar, congruent, SAS, ASA. They never mention all these things that you essentially memorize in your geometry class. And I know people who do the other way. And frankly, I'd rather hire the person who does well on the SAT.
Because that's the person who I think has the intuition. But anyway, we have to do this. And I probably shouldn't rant like that. And they tell us that angle 1 is congruent to angle 2.
So that and that are equal. All right, so already those look like alternate interior angles. If this line were parallel. In fact, I think that's enough to show that this line is parallel to this line. Because, if you view this as a transversal, if you view DC as a transversal, then you see that's a transversal between these two lines. And because the alternate interior angles are the same, or they're congruent, you know that those are going to be parallel lines.
But anyway, I don't know if that's at all useful. What are they going to ask us? So let's see, so I didn't have to even say those are parallel lines. So what do they tell us?