Download presentation

Presentation is loading. Please wait.

1
**Tangent Circles Finding solutions with both**

Law of Cosines and Stewart’s Theorem Inspired by (though not actually stolen from) problem #10 on the 2011 High School Purple Comet Competition With thanks to Luke Shimanuki for sharing his solution

2
3 inches

3
**AB = AC = BC = AP = BP = CP = 3 2 1 r + 1 r + 2 3 - r**

Find the following segments either as whole numbers or in terms of r (the radius of circle P) AB = AC = BC = AP = BP = CP = 3 2 1 r + 1 r + 2 3 - r

4
**AB = AC = BC = AP = BP = CP = 3 2 1 r + 1 r + 2 3 - r**

Let us now focus on two of the triangles – APC and APB I will redraw and enlarge the two triangles on the next slide

5
r + 1 3 - r r + 2 x 2 1 We will begin by using Law of Cosines with triangle APC to solve for angle PAC (x degrees)

6
**Law of Cosines a2 = b2 + c2 – 2bcCos A Triangle PAC 2 1 r + 1 r + 2**

x Law of Cosines a2 = b2 + c2 – 2bcCos A Triangle PAC

7
**Now we will use law of cosines a second time**

2 1 r + 1 r + 2 3 - r x Law of Cosines Triangle PAC Now we will use law of cosines a second time This time we will use it on Triangle PAB

8
**Law of Cosines - Triangle PAB**

2 1 r + 1 r + 2 3 - r x Law of Cosines - Triangle PAB a2 = b2 + c2 – 2bcCos A

9
3 inches

10
**Allows you to calculate the length of a cevian.**

Stewart’s Theorem Allows you to calculate the length of a cevian. A cevian is a line segment that extends from a vertex of a polygon to it’s opposite side. Medians and altitudes are examples of “special” cevians in a triangle. The formula is:

11
**We can rewrite the formula in a way that is easier to remember …**

Stewart’s Theorem A cevian is a line segment that extends from a vertex of a polygon to it’s opposite side. We can rewrite the formula in a way that is easier to remember … A man and his dad put a bomb in the sink ….

12
**Let’s plug our values in and see what we get**

Stewart’s Theorem Let’s plug our values in and see what we get a = b = c = d = m = n = 3 r + 2 r + 1 3 – r 2 1

13
a Stewart’s Theorem a = b = c = d = m = n = 3 r + 2 r + 1 3 – r 2 1

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google