Hands-on math!

Washington State Math Olympiad
Hints and Solutions
2011 Grade 7 Geometry
Problem
Hint
1) In the diagram below, EFLM is a square. B is the midpoint of segment AC and I is the midpoint of segment JH. All interior angles are right angles. Segments AB and JI both have length 8 units and segments BE, EF, and FI all have the same length. The total perimeter is 128 units. What is the length of KL?
  1. There are 3 different segment lengths:
    1. The square side length, ML: There are ___ of these
    2. The shorter side, AB = 8: There are ____ of these
    3. The KL length: ___ of these
  2. Set up the equation for the perimeter and solve for KL in terms of ML:

    KL = __________________
  3. You don't know the length of ML, so try a few values of ML to narrow down the values for KL by plugging them into the equation you found:



  4. KL = _____
2) Jerry wants to recreate the drawing at right using a larger circle. A is the center of the circle and segment TC is congruent to segment TB. What is the fewest number of angles he needs to measure and still be able to compute the measure of all the angles in the drawing?
 "Congruent", in this case, means the segments TC and TB are the same length.
Notice that if you know angle TAC then you also know angle TAB because those 2 triangles have sides of the same length.
Together, with angle CAB, they add to _____ degrees.
From this information you should be able to find out how many measurements you need:






You need _____ measurements.

Problem
Hint
3) Here is a quick sketch that a student made of a diagram she saw in a book. The student was in a hurry and didn't have time to use a protractor while making the sketch but the student was sure the line segments were straight, the two right triangles are similar, and the labeled angle in the smaller triangle on the left was 32 degrees. What was the measurement of the angle that the student saw in the book, in degrees, of the obtuse angle just to the right of the large triangle?
  1. Angle A is ______.

  2. Angle ____ is the same measure as angle A because they are vertical with each other.

  3. Angle C is _____

  4. Angle D is the same measure as angle ____ because they, too, are similar. Angle D = ____ degrees.

  5. Finally, angle ? and angle D add to ______ degrees, so it is _____ degrees.
4) The figure represents a net that you cut out along the solid outside lines and then fold along the interior dotted lines. The length of the side of the square is 5 units. The isosceles triangles have height h. What is the smallest number h must be greater than to guarantee the net can be folded to make a pyramid?
  1. The figure to the right is a side view of the net after it has been folded into a pyramid.

  2. From this figure, you should be able to find the minimum value of the height = greater than _____
5) Two isosceles triangles are inscribed in a circle so that they share a common base, see example in diagram. Triangle HLJ contains the center of the circle, P, in its interior. Of the angles labeled 1, 2, 3, and 4 how many must be acute?
  1. Imagine point H moving such that HJ goes through the center, P
  2. With this geometry, all of the angles of triangles HLJ and HKJ must be acute.
  3. Moving H to the left, such that triangle HLJ contains P also requires that all the angles of HLJ remain acute.
  4. Therefore, _____ of the angles must be acute.