



 
Hi Steve, There is a solution to your problem that is easier than Stephen's solution to Angela's problem. The solution comes from the fact that an ellipse is the locus of all points in the plane such that the sum of the distances from each of the points to two fixed points is a constant. This is classical mathematical language so, what does it mean? Drive two nails into a board. These are the fixed points called the foci. Cut a length of string longer than the distance between the foci. Tie the ends of the string to the nails. Place a pencil so the string forms a triangle with the pencil and the two nails the three vertices. Move the pencil, keeping the string taught and the curve traced by the pencil is an ellipse. At each point on the curve the sum of the distances from the pencil to the foci is a constant, the length of the string. To draw the ellipse you want you need to know the length of the string and the position of the foci. I redrew the ellipse and labeled some points. The point $C$ is the midpoint of the line segment $PQ.$ When the pencil is at the point $Q$ the string goes from $F_1$ to $Q$ and then back to $F_{2}.$ The distance from $Q$ to $F_2$ is equal to the distance from $P$ to $F_1$ and hence the length of the string is $F_1 Q + QF_2 = F_1 Q + PF_1 = PQ$ which in your case is 22 feet. When the pencil is at the point $R$ in the diagram the string goes from $F_1$ to $R$ and then from $R$ to $F_{2}.$ By the symmetry of the diagram the distances $F_1 R$ and $R F_2$ are equal and hence $R F_2 = 11$ feet. You also know that $RC = 9$ feet. Suppose the distance from $C$ to $F_2$ is $x$ feet. Triangle $RCF_2$ is a right triangle and hence from Pythagoras Theorem \[x^2 + RC^2 = RF_2^2 \mbox{ or } x^2 + 9^2 = 11^{2}.\] Solving for x gives $x = 6.32$ feet which is very close to 6' 4". Thus to draw your ellipse locate the midpoint $C$ of the straight side, measure 6" 4" each way to locate $F_1$ and $F_{2}.$ Pound a nail at these two points and tie a string between $F_1$ to $F_2$ so that the length of the string between the foci is 22 feet. Draw your half ellipse. I hope this helps,  


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