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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

1. Two lines \ell and m intersect at O at an angle of 28^\circ. Let A be a point inside the acute angle formed by \ell and m. Let B and C be the reflections of A in lines \ell and m, respectively. Find the number of degrees in \angle BAC.

2.Equilateral triangle ABC has centroid G. Triangle A'B'C' is the image of triangle ABC upon a dilation with center G and scale factor -2/3. Let K be the area of the region that is within both triangles. Find K/[ABC].

3.A sphere with radius 3 is inscribed in a conical frustum of slant height 10. (The sphere is tangent to both bases and the side of the frustum.) Find the volume of the frustum.

4.In tetrahedron ABCD, \angle ADB = \angle ADC = \angle BDC = 90^\circ. Let a = AD, b = BD, and c = CD.

(a) Find the circumradius of tetrahedron ABCD in terms of a, b, and c. (The circumradius of a tetrahedron is the radius of the sphere that passes through all 4 vertices of the tetrahedron.)

(b) Let O be the circumcenter of tetrahedron ABCD. Prove that \overline{OD} passes through the centroid of triangle ABC.

(The circumradius of a tetrahedron is the radius of the sphere that passes through all four vertices, and the circumcenter is the center of this sphere.)

*Last edited by mathstudent2000 (2013-08-12 12:56:28)*

Genius is one percent inspiration and ninety-nine percent perspiration

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,704

hi

I've got to out soon so I'll just look at Q1 for the moment.

See diagram. What is the size of the angles at E and F

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

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

thanks, i solved the questions

Genius is one percent inspiration and ninety-nine percent perspiration

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,704

Oh good. That's saved me a job then.

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

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**RandomPieKevin****Member**- Registered: 2015-07-02
- Posts: 29

Can you help me with Q3?

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 7,704

hi RandomPieKevin

Q3 has recently been asked by someone else and solved in another thread.

http://www.mathisfunforum.com/viewtopic.php?id=21271

championmathgirl's method for finding x is much better than mine! see post 11.

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

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**RandomPieKevin****Member**- Registered: 2015-07-02
- Posts: 29

@bobbundy

I posted something there. Can you help me there? Thanks.

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