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#1 2013-08-12 12:36:33

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

geometry problems

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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#2 2013-08-12 18:44:21

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,071

Re: geometry problems

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

View Image: mathstu1.gif

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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#3 2013-08-13 15:52:29

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

Re: geometry problems

thanks, i solved the questions


Genius is one percent inspiration and ninety-nine percent perspiration

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#4 2013-08-13 18:40:21

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,071

Re: geometry problems

Oh good.  smile  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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#5 2015-07-16 08:50:19

RandomPieKevin
Member
Registered: 2015-07-02
Posts: 28

Re: geometry problems

Can you help me with Q3?

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#6 2015-07-17 21:32:43

bob bundy
Moderator
Registered: 2010-06-20
Posts: 7,071

Re: geometry problems

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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#7 2015-07-18 05:56:59

RandomPieKevin
Member
Registered: 2015-07-02
Posts: 28

Re: geometry problems

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

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