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Maths, maths and moar maths. Help please? (1 Viewer)

MiseryParade

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My teacher refused to solve this question for me, because it isn't being tested for our assessment task and there wasn't enough time for him to explain. I'm guessing part b is doable after part a has been solved, but I've no idea how to solve part a. Help? ;_;

a) Show that every circle that passes through the intersections of the circle x^2 + y^2 = 2 and the straight line y=x can be written in the form (x-μ)^2 + (y+μ)^2 = 2(1+μ^2).

b) Hence find the equation of such a circle which also passes through (4, -1).


+ question 2!

Suppose that a regular polygon has 'n' sides of length 1.

a) What will be the length of the side of the regular polygon with 2n sides that is formed by cutting off the vertices of the given polygon?

b) Confirm your answer in the case of:
i. cutting the corners off an equilateral triangle to form a regular hexagon
ii. Cutting the corners off a square to form a regular octagon


last question that isn't really a question: what abbreviations are you allowed to use in senior school? Teachers want us to write opposite instead of opp and straight instead of str, but are you allowed to use the angle symbol and the = symbol to write a proof?
 

theind1996

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My teacher refused to solve this question for me, because it isn't being tested for our assessment task and there wasn't enough time for him to explain. I'm guessing part b is doable after part a has been solved, but I've no idea how to solve part a. Help? ;_;

a) Show that every circle that passes through the intersections of the circle x^2 + y^2 = 2 and the straight line y=x can be written in the form (x-μ)^2 + (y+μ)^2 = 2(1+μ^2).

b) Hence find the equation of such a circle which also passes through (4, -1).


+ question 2!

Suppose that a regular polygon has 'n' sides of length 1.

a) What will be the length of the side of the regular polygon with 2n sides that is formed by cutting off the vertices of the given polygon?

b) Confirm your answer in the case of:
i. cutting the corners off an equilateral triangle to form a regular hexagon
ii. Cutting the corners off a square to form a regular octagon


last question that isn't really a question: what abbreviations are you allowed to use in senior school? Teachers want us to write opposite instead of opp and straight instead of str, but are you allowed to use the angle symbol and the = symbol to write a proof?
I'm not sure, but I have a feeling that you need to use the perpendicular distance formula for a). Try that out.

Oh, and you CANNOT use abbreviations for opp. and alt. etc., but you CAN use < for angle and = for equal etc.

Your teachers are telling you the right thing regarding all of that.
 
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MiseryParade

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I'm not sure, but I have a feeling that you need to use the perpendicular distance formula for a). Try that out.

Oh, and you CANNOT use abbreviations for opp. and alt. etc., but you CAN use < for angle and = for equal etc.

Your teachers are telling you the right thing regarding all of that.
Tried everything. How do these figures morph into that funky micro symbol...? ;_;

Can you use // for parallel then? How about...//logram? (I've used this 7-10 and no one bitched about it)
 

Timske

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Find points of intersection which is (1,1) and (-1,-1)

If you expand this and simplify (x-μ)^2 + (y+μ)^2 = 2(1+μ^2). You will get x^2 + y^2 = 2
 

theind1996

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Tried everything. How do these figures morph into that funky micro symbol...? ;_;

Can you use // for parallel then? How about...//logram? (I've used this 7-10 and no one bitched about it)
Look at this sheet that I've attached.

I'd imagine Ruse would treat the abbreviations the same as your school.
 

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Carrotsticks

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Solve the line and circle simultaneously. You will get two points (1,1) and (-1,-1) and the family of circles must pass through those two points (we have infinite circles doing so)

1. The most general equation of a circle is (x-a)^2 + (y-b)^2 = c^2. Sub in the two points and you will get a pair of simultaneous equations with 3 variables.

2. Find an expression for a and b in terms of c.

3. Sub those in and simplify and you should get the required result.

----------------------------------------

The next question is a little ambiguous because it didn't specify how far we cut into the vertices, unless they want us to do so until we get another regular polygon.
 

MiseryParade

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Solve the line and circle simultaneously. You will get two points (1,1) and (-1,-1) and the family of circles must pass through those two points (we have infinite circles doing so)

1. The most general equation of a circle is (x-a)^2 + (y-b)^2 = c^2. Sub in the two points and you will get a pair of simultaneous equations with 3 variables.

2. Find an expression for a and b in terms of c.

3. Sub those in and simplify and you should get the required result.

----------------------------------------

The next question is a little ambiguous because it didn't specify how far we cut into the vertices, unless they want us to do so until we get another regular polygon.
thank you!
 

MiseryParade

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^didn't think it'd belong here, thanks carrot ;P

Extra help anyone? D:

The ratio of the interior to the exterior angles of a regular polygon is 4 : 1
a) Find the size of each exterior angle
b) How many sides does the polygon have?

The answers were 24 degrees and 15 sides respectively, but I just can't get that. I keep getting 36 degrees and 10 sides... ;_;
 

Carrotsticks

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^didn't think it'd belong here, thanks carrot ;P

Extra help anyone? D:

The ratio of the interior to the exterior angles of a regular polygon is 4 : 1
a) Find the size of each exterior angle
b) How many sides does the polygon have?

The answers were 24 degrees and 15 sides respectively, but I just can't get that. I keep getting 36 degrees and 10 sides... ;_;
 

Carrotsticks

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There's also a faster way of doing it.

Since the Interior and Exterior angles are Supplementary and in the ratio 4:1, we can say that the size of the Exterior angle is simply 180/5 = 36 degrees and subbing that into our formula for the Exterior angles, it follows immediately that N = 10 (that rhymes!).
 

Drongoski

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The answers were 24 degrees and 15 sides respectively, but I just can't get that. I keep getting 36 degrees and 10 sides... ;_;
I got the same answer as did Carrot. So I think your answer is correct.
 
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MiseryParade

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Thanks Drongoski and carrot (again)! There were so many wrong answers in this past paper - I was inches away from a huge panic attack ):
 

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