Problem Sheet 1 MT1100 From Euclid to Mandelbrot To be returned on Thursday 18 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. This question revises material you should already know on congruent triangles. If you've forgotten this material, look it up! (i) Write down what it means for two triangles to be congruent. (ii) Write down the statements of the SSS, SAS, ASA, AAS conditions that show two triangles are congruent. (iii) Write down an example of two triangles that are not congruent, but satisfy the SSA con- dition: two pairs of sides are equal in length and a corresponding non-included angle is equal. 2. Suppose we are given points A and B such that |AB| = 1. Construct, using a ruler and compass, a point C such that |AC| = 3. [I'm asking for a diagram, a list of steps in the construction, and a justification.] 3. In the diagram below, |AB1| = 1, and |B1B2| = |B2B3| = |B3B4| = |B4B5| = 1. Also, all the angles AB¡Bi+1 are right angles. (i) Show that | AB5| = V5. (ii) Show that, given A and B1, we can construct the point B5 (by ruler and compass). [Just a list of steps will do. No justification required, and only include a diagram if it helps make things clearer.] 1 B2 1 B3 0 BA 1 B5 1 B1 1 A 4. Suppose that ABCD below is a parallelogram (so the lines AB and CD are parallel, and the lines AD and BC are parallel). (i) Some pairs of the angles in the diagram are equal. Which pairs are they, and why do you know they are equal? (ii) The triangles ABC and CDA are congruent. Which rule can you use to show this? (iii) Show that |AB| = |CD| and | AD| = |BC|. [This question proves Theorem 1.2 in the notes.] 2
B d C e f C b a A D 3