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An easier solution of a Diophantine problem about triangles, - download pdf or read online

By Euler L.

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Graph the ellipse when k = 3. 16. For what values of k is this the equation of a circle? 17. Graph the following hyperbola and find its asymptotes: 2 6 2 4 X 56 CHAPTER TEST 5E 1. Consider two line segments, one with endpoints (JL, 2). 4, 2\. Find the midpoint of the line segment whose endpoints are the midpoints of the two given line segments. 2. 1, 1).. Let C be the midpoint of AB and let D be the midpoint of AC. If the coordinates of D are (2, 3/2), find the coordinates of B. Problems 3-5 concern a circle with a point P, coordinates (0, 0 ) , inside.

21 17. log (6 - x) = 2 18. log (log x) = 1 - 3 * 2 46 CHAPTER TEST 4E In Problems 1 and 2, if log 3 = x and log 5 = y evaluate the given logarithm in terms of x and y. (I) 1. log 2. log [(45)(75) ] In Problems 3-6 simplify 3. log8(l) 4. log (42) 2 5. © 6. 1 0 , ^ 6 ) In Problems 7 and 8 write each expression as a single logarithm. 7. 8. In Problems 9 and 10 use logarithms to evaluate the given expression. 9. 8) 10. In Problems 11 and 12 simplify. 11. log /(9x 12 ln « 13. + 4) 21(4a: + 1) 5 U ( Ä + 2)j Draw the graph of fix) = (—1 47 CHAPTER TEST 4E 14.

In Problems 6-8 the point P with coordinates (3, 1 ) , is on the circle in question. 6. Find the equation of the circle with center (-1, 2 ) . 7. If P is one endpoint of a diameter of the circle and Q is the other endpoint, find the coordinates of Q. 8. Choose a point P on the circle, different from P and Q. Prove that PQR is a right triangle. , 0,{{a, (c, 9. b)}, {circle of radius r with center d)}). x2 10. 2x2 11. x2 + y2 + 4x - Qy + 7 = 0 + 2y2 + y2 - \2x - 20y + 75 = 0 + 8x - Qy + 32 = 0 In Problems 12 and 13, given the equation of the parabola, find the vertex, the axis, and the direction the parabola opens.

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An easier solution of a Diophantine problem about triangles, in which those lines from the vertices which bisect the opposite sides may be expressed rationally by Euler L.


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