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    Why does Prof. Loh multiply r minus d with d plus r?

    Module 2 Day 7 Challenge Part 5
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    • The Blade DancerT
      The Blade Dancer M0★ M1★ M2★ M3★ M4 M5
      last edited by

      Do angles have to do with the length of a line segment in contrast to another line segment altogether in contrast to another two sets of line segments? Like for example PA to PB vs PC to PD. Do angles determine the ratios?

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

      debbieD 1 Reply Last reply Reply Quote 2
      • debbieD
        debbie ADMIN M0★ M1 M5 @The Blade Dancer
        last edited by debbie

        @TSS-Graviser It's kind of neat that all of your questions about angles, arcs and Power of a Point Theorem are tying in together! Now I see that you're trying to synthesize it in your head in order to make it into a tool for yourself to use. It's very good to ask, "When exactly can I use this? When can I not use this?" If you don't mind, I'll take the long approach again and backtrack to the proof of Power of a Point and how it all ties in together.

        First, let's start with the two chords that intersect in a circle.
        The yellow angle inscribes the yellow arc here:

        M2W2-what-about-arcs-and-angles.png

        Here, with the second chord, we can connect its angle vertex in such a way that the same arc is inscribed by the second yellow angle.

        M2W2-what-about-arcs-and-angles2.png

        Now it's time to look at the other angles of the triangles! There are two other angles that inscribe the same arc of the circle, the green ones shown here:

        M2W2-what-about-arcs-and-angles3.png

        And we also know that the angles formed by the "X" are the same, so the two triangles drawn inside the circle are similar! (They're not necessarily congruent, because we don't know any of the lengths.)

        These similar triangles are analogous to the similar triangles that we saw in the Day 7 Challenge video:

        781c6c19-6651-4d17-98ee-6978763eca58-image.png

        The sides \( \overline{PA}\) and \(\overline{PD}\) are matching sides that go from \( \angle \text{yellow} \) to \( \angle \text{green} \), and the sides \( \overline{PB} \) and \(\overline{PC}\) are matching sides that go from \( \angle \text{yellow} \) to the third angle (not colored). This is the reason for writing the ratios

        $$ \frac{ PA}{PD} = \frac{PB}{PC} $$

        If this looks confusing, perhaps try to imagine that the large triangle is twice the dimensions of the small triangle. Then \( \overline{PA}\) would be double \( \overline{PD},\) and \(\overline{PB}\) would be double \(\overline{PC}.\) Then

        $$ \frac{ PA}{PD} = \frac{PB}{PC} = 2 $$

        At any rate, if you multiply both sides by \( \overline{PD} \) and \( \overline{PC},\) you can cancel the denominators to get the Power of a Point theorem:

        $$ PA \times PC = PD \times PB $$

        What's the pattern that we can find in this formula? Well, \( \overline{PA}\) and \(\overline{PC}\) are on the same chord, \( \overline{CA},\) and \(\overline{PD}\) and \(\overline{PB}\) are on the same chord \(\overline{BD}.\) In other words, try to look for the entire chords in the picture; their segments will be the ones that you multiply together in Power of a Point. The purple line below is one chord:

        M2W2-what-about-arcs-and-angles-purple-line.png

        And this purple line is another:

        M2W2-what-about-arcs-and-angles-purple-line2.png

        Now, here is the tool we have for applying the Power of a Point Theorem:

        • Take the two segments of one chord, and multiply them together.

        • Take the two segments of the other chord, and multiply them together.

        • These two products should equal each other.

        I hope this made sense! Let me know! 🙂

        1 Reply Last reply Reply Quote 2
        • The Blade DancerT
          The Blade Dancer M0★ M1★ M2★ M3★ M4 M5
          last edited by

          Sorry I get the first part now but I don't get all those angle equations in the second half. I just want to know if angles determine side length relationships.

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

          debbieD 1 Reply Last reply Reply Quote 1
          • debbieD
            debbie ADMIN M0★ M1 M5 @The Blade Dancer
            last edited by

            @TSS-Graviser Oh, are you referring to the equations like

            $$ \frac{PA}{PD} = \frac{PB}{PC}?$$

            I think I might see what you're confused about. Are you thinking that \(PA, PD, PB,\) and \(PC\) are measures of angles? They are actually lengths of lines segments.

            If we were talking about the measure of angles, we would write this as

            $$ \angle CPD $$

            for the yellow angle, for example.

            By the way, if you want to write this yourself, you can type

            \\( \angle CPD \\)

            🙂

            1 Reply Last reply Reply Quote 1
            • The Blade DancerT
              The Blade Dancer M0★ M1★ M2★ M3★ M4 M5
              last edited by

              No I mean all that matching stuff and whatnot. Is there like a simplified explanation that like I can digest?

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

              debbieD 2 Replies Last reply Reply Quote 2
              • The Blade DancerT
                The Blade Dancer M0★ M1★ M2★ M3★ M4 M5
                last edited by

                And by the way, when can power of a point be used? It seems that I can use it anywhere with intersections, but is that always the case?

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

                1 Reply Last reply Reply Quote 1
                • debbieD
                  debbie ADMIN M0★ M1 M5 @The Blade Dancer
                  last edited by

                  @TSS-Graviser Oh, I see! Do you mean that you're not sure how to figure out which sides match? Try imagining that you are folding the screen like closing a book, so that the yellow angles lie on top of each other and the green angles also lie on top of each other. Do you see that \( \overline{PA}\) will lie right on top of \(\overline{PD}?\)

                  debbieD 1 Reply Last reply Reply Quote 1
                  • debbieD
                    debbie ADMIN M0★ M1 M5 @debbie
                    last edited by

                    @debbie Yes! Whenever you have two chords intersecting inside a circle, you can use Power of a Point. 🙂

                    1 Reply Last reply Reply Quote 1
                    • debbieD
                      debbie ADMIN M0★ M1 M5 @The Blade Dancer
                      last edited by

                      @TSS-Graviser The simplified version was found at the bottom of the earlier very long reply:

                      Now, here is the tool we have for applying the Power of a Point Theorem:

                      Take the lengths of two segments of one chord, and multiply them together.

                      Take the lengths of two segments of the other chord, and multiply them together.

                      These two products should equal each other.

                      I hope this made sense! Let me know! 🙂

                      1 Reply Last reply Reply Quote 2
                      • The Blade DancerT
                        The Blade Dancer M0★ M1★ M2★ M3★ M4 M5
                        last edited by

                        Oh ok thanks. What I've been trying to say is that above in the pictures with the arcs, right, how do all these line segments on the same chord mean anything? And what's the point of labeling the orange arc and the green arc? Do they have a purpose?

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

                        debbieD 1 Reply Last reply Reply Quote 1
                        • debbieD
                          debbie ADMIN M0★ M1 M5 @The Blade Dancer
                          last edited by debbie

                          @TSS-Graviser That's a good question; it depends on what you mean by meaningful! The line segments happen to be the side lengths of the triangles in the picture. Maybe you will see a math question that asks for one of the lengths of the triangles. In that case, knowing Power of a Point will be very meaningful!

                          The yellow arc goes with the yellow angle. The yellow angle is inscribed inside the yellow arc. That means that the yellow angle's measure is \( \frac{1}{2}\) of the measure of the arc.

                          So to answer your emailed question, 🙂 , suppose an angle opens up to exactly half of the circle. That half of the circle is an arc with measure \( 180^{\circ}.\) That means the angle is \( 90^{\circ}.\)

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