Problem Sheet 3 MT1100 From Euclid to Mandelbrot Hand in your solutions, with this coversheet, for feedback on Thursday 1 November 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. (This question shows that a perpendicular bisector is a line.) Let A, B, P, Q, R be points with R on the line PQ. (See the diagram below.) Suppose that | AP| = |BP| and | AQ| = |BQ|. Show that | AR| = |BR]. [Hints: Find some triangles that are congruent. Start by showing the angles a and B are equal.] P ? A B R Q 2. Given points A and B, construct six points that form the corners of a regular hexagon (a regular 6-sided polygon). Just give a diagram. The steps of the construction are not required; neither is a justification. Hint: you do not need to use a ruler at any point! 3. (This question fills in the missing half of the proof of Theorem 1.3.) Suppose that p and q are constructible lengths. Show that, given points A= (0, 0) and B= (1,0), we can construct the four points (p,q), (-p,q), (p,-q) and (-p,-q). Just give a diagram and the steps of the construction. No other justification is required. 4. Suppose you are given a triangle ABC with the angle ACB obtuse (greater than T/2). Construct (using a ruler and compass) a right-angled triangle PAB with the same area as ABC, and with the right-angle at P. [Hint: First construct a circle with diameter AB. Use the formula for the area of a triangle involving its height!] Give a justification for your construction. 2