Download Problem Set 4.1: Geometry

Problem Set 4.1: Geometry
Question 1: AMC10B 2013 Problem 7
Question 2: AMC10B 2014 Problem 15
Question 3: AMC10A 2014 Problem 23
Question 4: AMC10B 2014 Problem 19
Question 1: AMC10B 2013 Problem 7
(a) How many different triangles can you draw on this circle?
(b) What can you say about each one? What type are they? What angles do they
have?
(c) You will need Thales’ and Pythagoras’ Theorem for this question (and the rest
in this set!) - if you don’t know what they are, find out now!!
(d) Can you find the lengths of the sides for the scalene triangles you found in
part (a), and hence their area?
[Hint: this hexagon inscribed in a circle might help]
Question 2: AMC10B 2014 Problem 15
(a) What do you know about the angles at D?
(b) What do you know about triangle ADE? How about triangle ADF?
(c) What is the area of triangle EDF?
(d) Work out the ratio of lengths AE:EF and compare this with the ratio AD:DF –
what do you notice? How about the ratio of areas ADE:EDF?
EXT: Try bisecting the angles of other triangles and finding the ratio of sides and
areas like this – what do you notice?
Can you prove your result?
Question 3: AMC10A 2014 Problem 23
(a) Try drawing the piece of paper after the fold has been made. Does this tell
you anything about what is going on?
(b) Can you work out the length of the dotted line?
(c) Are there any (right-angled) triangles you can draw anywhere? Do they tell
you anything?
(d) Does this picture tell you anything?
(e) How about this one?
Question 4: AMC10B 2014 Problem 19
(a) Start by drawing a picture. Draw some chords between the two points. Which
chords are special in your drawing?
(b) Here is a chord that just touches the inner circle. What do you know about
this chord?
(c) Are there any triangles you can draw on this diagram that will give you some
more information?
(d) Here is a picture of two of these chords. Can you find the angle at D? What
does this tell us about this problem? [Hint: consider arcs of the circle]