Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2012 Grade 8 Geometry
Problem
1) Jodi looks at an analogue clock when it is exactly 11:00 am. The angle between the hands is 30 degrees. What time is it when the angle between the hands first hits 85 degrees? Note: both hands move at different constant rates.
2) The base of an isosceles triangle has vertices at (0,1) and (6,3) as shown. Its height is half its base. What can the coordinates of its vertex be?
3) Avery wants to create a net that he can cut out and fold to make a regular dodecahedron. A dodecahedron is made up of 12 pentagons. Three of them come together at a vertex. What is the left over angle, angle ABC, that will create the point at vertex B once it is folded? See diagram.

Problem
4) Rectangle PQRS is the image of rectangle ABCD after a rotation of 90 degrees counterclockwise (↶) about the origin followed by a translation 1 unit up and 1 unit to the right. Rectangle TUVW is the image of rectangle ABCD after a translation 1 unit up and 1 unit to the right followed by a rotation of 90 degrees counterclockwise about the origin. Clearly plot the set of points that is the intersection of rectangle PQRS with rectangle TUVW on the coordinate axes provided on the answer sheet.
5) In the drawing below, the two lines that look parallel are parallel. With the given information can you determine the measure of the angle labeled x? If so what is it? If not, which of the labeled angles, 1, 2, 3, or 4, do you need to know to determine x?