SOLUTION: Hello! I was having some trouble with one of my geometry homework problems. Here is the problem: In pyramid ABCP, point M is the midpoint of BC. The segment PM is perpendicular

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Question 1134517: Hello!
I was having some trouble with one of my geometry homework problems. Here is the problem:
In pyramid ABCP, point M is the midpoint of BC. The segment PM is perpendicular to the plane of the base and PM = 9 cm. The radius of the circle circumscribed around ∆ABC is 24 cm and m∠CAB = 30°. Find the length of edge PB.
Thanks!
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.

            I will give you only basic instructions, determining the direction and leaving the details to you. 


1.  Let O be the center of the circle circumscribed around the triangle ABC.

    Then the angle BOC is 60 degrees.

    The triangle BOC is ISOSCELES with two sides of 24 cm long (the radii) and the angle BOC of 60 degrees.

     Use the Cosine Law to find BC.



2.  As soon as you find BC, you know BM as half of BC.



3.  Then the triangle BMP is a right angled triangle, in which you know the legs.

    Then find the hypotenuse PB, which is what you need to find.


Happy calculations !


=======================


Later addition 

The triangle  BOC is ISOSCELES  with the angle  BOC  of 60 degrees between two equal sides ===============>


    hence, it is EQUILATERAL, and all its sides are 24 cm long.


So,  BC  is 24 cm;  hence,  BM is 12 cm (as half of BC).


Thus  |PB| = sqrt%2812%5E2+%2B+7%5E2%29 = sqrt%28144+%2B+49%29 = sqrt%28193%29.


ANSWER.  |PB| = sqrt%28193%29 cm = 13.89 cm (approximately).

Solved.


Thus I solved it  COMPLETELY  without leaving the room for you  (for your calculations . . . ).  Sorry for it . . .



Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Interesting problem. I solved it using trigonometry; the result suggested a solution using only geometry should be possible, so I went back and found one.

Let O be the center of the circumscribed circle, so OA=OB=OC=24.

We are told that angle CAB is 30 degrees. It is an angle inscribed in the circle. Angle COB is an central angle cutting off the same arc as angle CAB, so angle COB is 60 degrees.

Triangle COB is isosceles; M is the midpoint of the base. So triangles COM and BOM are congruent right triangles, with angles COM and BOM each 30 degrees (half of angle COB).

Now you have enough information to find the length of BM and CM.

Finally, angle BMP is a right angle, so you can find the length of PB using the Pythagorean Theorem.