Virginia Mathematics Teacher Spring 2017

So l ut i ons t o Fa l l 2016 HEXA Cha l l enge Probl ems

SOLUTION:

Let us evaluate each addend in the following expression individually:

March In an Isosceles triangle ABC, AB=BC, Angle B is 20 degrees, AC is five units Point D is on BC so that Angle BAD is 30degrees, and angle DAC is 50 degrees. Point E is on the side AB, so that Angle ECB is 60 degrees and angle ECA is 20 degrees. Find the length of DE. See Figure 1. SOLUTION: First, in the triangle ADC, the measure of angle D is 50 degrees (sum of the angles in a triangle equals 180 degrees). This means that triangle ADC is isosceles. Therefore side DC measures 5 units. In the trian- gle EAC, the measure of angle E is 80 degrees. This means that triangle EAC is also isosceles, and the measure of side EC is also 5 units. In the isosceles DCE, the base angles E and D are congruent. Because the third angle, DCE, of the triangle is 60 degrees, angles D and E must also measure 60 degrees. There- fore, triangle DCE is equilateral, which means that side DE is 5 units. (see Figure 2)

Virginia Mathematics Teacher vol. 43, no. 2

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