Math Is Fun Forum

Kage Suzuki

Guest

Geometry help

1. ABC and CDE are equilateral triangles with side lengths 3 and 5, respectively. Find the perimeter of 4ADE.
Picture:

2. X,Y and Z are three points on a unit sphere (i.e. a sphere with radius 1). The space distance between any pair of the three points is √2 (i.e. XY = YZ = ZX =√2). An ant can only crawl on the surface of the sphere. The ant starts from point X, passes through points Y and Z, and then returns to X. What is the shortest possible distance of this trip?
Picture:

Thank you so much!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,580

Re: Geometry help

hi Kage Suzuki

Welcome to the forum.

1)  DB = DC - BC = 5 - 3 = 2

Angle ABC = 60, so angle ABD is 120.  So you can use the cosine rule
https://www.mathsisfun.com/algebra/trig-cosine-law.html
to calculate AD.

2) On the Earth (but it's also true on any sphere) the shortest route around the surface of the Earth between two points is called the great circle route.  If you are looking at a flat map of the world and you see an aircraft route from say London to New York you might be puzzled as to why the route seems curved.  It's an effect of the map projection.  If you plotted the same route on a globe you'd see it is the shortest. Great circles are centred on the centre of the Earth and have radius = the radius of the Earth.

So the radius of the curve that the ant takes is r=1, but what is the angle at the centre of the sphere in gong from X to Y.  If you draw the triangle XOY where O is the centre of the sphere, we have OX = OY = 1 and XY = root(2).  So the triangle is isosceles.  Split it in half to make a right angled triangle and we have 1 and root(2) / 2.  By Pythagoras the height is also root(2) / 2 so the angle at the centre for the half triangle is 45.  So angle XOY = 90. 

So the curved great circle distance is 1/4 of the circumference. YZ and ZX are the same so multiply by 3 for the whole distance.

Hope that helps,

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Kage Suzuki

Member

Registered: 2021-05-26

Posts: 4

Re: Geometry help

Thank you for the help!

For the first one, I managed to solve for what AD is, but I'm stuck on getting the value of AE. I was thinking maybe I could use similar triangles (pretend the point at which DC and AE intersect is called O) DOE and AOC to solve for side AE, but I've been trying and I haven't been able to do that. Could you help?

For the second one, I got

, is that right? I found the circumference of the middle of the sphere to be just pi, and 1/4 of that is 1/4pi, and then you multiply that by 3.

Thanks in advance!

Last edited by Kage Suzuki (2021-05-26 02:09:04)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,580

Re: Geometry help

hi

I see you've decided to become a member.  Good move I think!

Sorry I missed that part.  I suppose you could try similar triangles, but why not the cosine rule again.  Angle ACE is 120 again so you can get AE in triangle ACE.

As the radius = 1 the circumference is 2pi.r = 2 pi The method after that is correct.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Kage Suzuki

Member

Registered: 2021-05-26

Posts: 4

Re: Geometry help

I got it, thank you so much!