• Categories
  • Recent
  • Tags
  • Popular
  • Users
  • Groups
  • Login
Forum — Daily Challenge
  • Categories
  • Recent
  • Tags
  • Popular
  • Users
  • Groups
  • Login

How do we know that any other corresponding sides will multiply by 9/15?

Module 2 Day 5 Challenge Part 2
3
4
39
Loading More Posts
  • Oldest to Newest
  • Newest to Oldest
  • Most Votes
Reply
  • Reply as topic
Log in to reply
This topic has been deleted. Only users with topic management privileges can see it.
  • T
    The Blade Dancer M0★ M1★ M2★ M3★ M4 M5
    last edited by debbie Dec 20, 2020, 7:21 PM Dec 20, 2020, 1:12 AM

    Module 2 Day 5 Challenge Part 2

    Question is in the title ^

    The Blade Dancer
    League of Legends, Valorant: Harlem Charades (#NA1)
    Discord: Change nickname if gay#7585

    D 1 Reply Last reply Dec 20, 2020, 7:10 PM Reply Quote 5
    • D
      debbie ADMIN M0★ M1 M5 @The Blade Dancer
      last edited by debbie Dec 20, 2020, 7:20 PM Dec 20, 2020, 7:10 PM

      @the-blade-dancer Hi again! 🙂 Oh, nice. I think you're referring to the similar triangles △PCD\bigtriangleup PCD△PCD and △PAB\bigtriangleup PAB△PAB inscribed within the trapezoid below.

      M2D5-forum-xiao-ma-why-9-over-15.png

      How do we know that these triangles are similar? Well, ∠PCD\angle \textcolor{blue}{PCD}∠PCD and ∠PAB\angle \textcolor{blue}{PAB}∠PAB are equal because of the parallel lines CD‾\overline{CD}CD and AB‾\overline{AB}AB which are intersected by the transversal AC‾.\overline{AC}.AC. The other transversal line, BD‾,\overline{BD},BD, which crosses the same parallel lines, creates equal angles ∠PDC\angle \textcolor{magenta}{PDC}∠PDC and ∠PBA.\angle \textcolor{magenta}{PBA}.∠PBA. And since ∠BPA\angle \textcolor{turquoise}{BPA}∠BPA and ∠DPC\angle \textcolor{turquoise}{DPC}∠DPC are formed from the "X" made by BD‾\overline{BD}BD crossing AC‾,\overline{AC},AC, they are also equal. △PCD\bigtriangleup PCD△PCD and △PAB\bigtriangleup PAB△PAB have the same angles, so they are similar!

      It's a little easier to see this if you rotate △PCD\bigtriangleup PCD△PCD around so that it's oriented in the same way as △PAB.\bigtriangleup PAB.△PAB.

      M2D5-forum-xiao-ma-why-9-over-15-similar-triangles.png

      We can shrink down both triangles by dividing their dimensions by 15.15.15. This makes the longest side of △PAB\bigtriangleup PAB△PAB only 111 long. It's like a reference triangles. Do you see that the longest side of △PCD\bigtriangleup PCD△PCD is now 915?\frac{9}{15}?159​?

      M2D5-forum-xiao-ma-why-9-over-15-shrunken-down-triangle.png

      Each of the sides of the small triangle are 915\frac{9}{15}159​ the length of their corresponding side on the large triangle. (This ratio actually simplifies to 35.\frac{3}{5}.53​.) To get the ratio of their areas, you square the ratio of their sides. That's just a complicated way of saying the larger triangle has area (35)2\left( \frac{3}{5} \right)^2(53​)2 the area of the smaller triangle. 🙂

      For more examples about similar triangles, you can read this post by @thomas 🙂 !

      Good luck on the rest of this lesson, and thanks again for asking this question!

      @thomas said in Why is the ratio of △PCD to △PAB the square of the ratio of their sides?:

      Hi tidyboar,

      Good question! This is actually part of a general result in geometry: If I have two shapes that are similar (so that one is just a scaled-up version of the other), and the "scale factor" is rrr, then the ratio between their areas is r2r^2r2. Here is a picture with some examples:

      669a3948-7d46-4a0c-add2-1ad301dd7670-image.png

      As you can see, it's even true when the shape looks really weird!

      To explain why this is true, I think it's best to think about the square example. For squares, it's pretty easy: If the first square has side length aaa, and the second square has side length ararar (so it's rrr times bigger), then their areas are a2a^2a2 and a2r2a^2r^2a2r2, the ratio between the areas will be a2:a2r2a^2:a^2r^2a2:a2r2, which is just 1:r21:r^21:r2!

      So what about a triangle? What if the sides of one triangle are a,b,ca,b,ca,b,c, and the sides of the scaled-up triangle are ar,br,crar,br,crar,br,cr?

      The key idea is that when you scale-up a triangle, then everything inside the triangle is scaled up by the same factor. So, let's say the base of the triangle is bbb and the height is hhh. Then, the base of the scaled-up triangle must be brbrbr, and the height must be hrhrhr:

      fa86ca82-1c8d-42fa-a62f-768685e4979e-image.png

      What's the area ratio now? Well, the first triangle has area 12bh\frac12bh21​bh, and the second triangle has area 12(br)(hr)=12bhr2\frac12(br)(hr)=\frac12bhr^221​(br)(hr)=21​bhr2. That means that the ratio between their two areas is 12bh:12bhr2\frac12bh:\frac12bhr^221​bh:21​bhr2, which is the same ratio as 1:r21:r^21:r2. Ta-da!

      You're probably wondering how you can prove this for all possible shapes. It's an interesting question to think about! See if you can come up with any good explanation as to why it would work for every shape you could draw.

      Now, to finally answer the original question, we know that the triangles △PCD\triangle PCD△PCD and △PAD\triangle PAD△PAD are similar, and the ratio of their sides is 9:159:159:15. That means that △PAD\triangle PAD△PAD is 159\frac{15}{9}915​ times bigger, or 53\frac{5}{3}35​. So, by what we discussed, its area must be 259\frac{25}{9}925​ bigger, and that's why the ratio of the areas is 9:259:259:25.

      So in general, if you have two similar triangles with corresponding sides of ratio a:ba:ba:b, then the ratio between the areas will be a2:b2a^2:b^2a2:b2.

      I hoped that helped. Be sure to let us know if you have more questions. Happy learning!

      Thomas

      1 Reply Last reply Reply Quote 5
      • B
        Bulba_Bulbasaur M0★ M1★ M2★ M3
        last edited by Dec 20, 2020, 11:08 PM

        Just wondering, how long does it take for you to make a single post? You seem to work a lot.

        revenge of the math nerds!

        D 1 Reply Last reply Dec 20, 2020, 11:41 PM Reply Quote 4
        • D
          debbie ADMIN M0★ M1 M5 @Bulba_Bulbasaur
          last edited by Dec 20, 2020, 11:41 PM

          @bulba_bulbasaur 🙂 I think maybe I spent 30 minutes on this particular post, most of which was spent making the diagrams. I really like making diagrams because, like the saying goes, "A picture is worth a thousand words..." 🖼 🖼

          1 Reply Last reply Reply Quote 3

          • 1 / 1
          1 / 1
          • First post
            3/4
            Last post
          Daily Challenge | Terms | COPPA