Problem Sheet 2 MT1100 From Euclid to Mandelbrot Hand in your solutions, with this coversheet, for feedback on Thursday 25 October 2018 YOUR FEEDBACK TO THE LECTURER (e.g., what was hard, interesting, fun, etc. this week): FEEDBACK ON YOUR SOLUTIONS FROM THE MARKER: 1
1. Let C1 be the circle with centre (0, 0) and radius 2. Let C2 be the circle with centre (1, 2) and radius 3. Find the points of intersection of the circles C1 and C2. 2. Let a be a fixed real number, and let L be the line through (0, 0) and (1, a). Let C be the circle with centre (7, -1) and radius 5. For which value(s) of a do L and C intersect in exactly one point? (Justify your answer.) Draw a diagram illustrating a case when L and C intersect in one point. For which values of a do L and C intersect in two points? In zero points? 3. Let A and B be distinct points. In the diagram below, ABDC is a rectangle (which can be any height you choose), G bisects AB, and EF is the perpendicular to AB dropped from E. C D E A F 0 G B (i) Write down a list of ruler and compass steps (you can quote constructions from the lectures for some steps) to show that F can be constructed from A and B using a ruler and compass. (ii) Show that |AF| = |AB|/3. Hint: As a first step, note that the triangles AEF and ADB are similar. Then find another pair of similar triangles. [The diagram, together with the list of steps in (i) and the justifiction in (ii) shows that trisecting a length using a ruler and compass is possible. So (ii) provides a proof of correctness for this construction.] 4. (i) Show that, given distinct points A and B, you can construct a point C so that ABC is an equilateral triangle. [Give a diagram and a list of steps; do not give a justification.] (ii) Show that, given distinct points A and B, you can construct points X and Y so that ABXY is a square. [Give a diagram and a list of steps; do not give a justification.] 2