# Michelle E. Chung

* EMAT6680 Assignment 6: Triangle of Medians

2. If the original triangle is equilateral, then the triangle of medians is equilateral.
Will an isosceles original triangle generate an isosceles triangle of medians?
Will a right triangle always generate a right triangle of medians?
What if the medians triangle is a right triangle?
Under what conditions will the original triangle and the medians triangle both be right triangle?

1) Medians triangle of an equilateral triangle

Fact A medians triangle of an equilateral triangle is an equilateral triangle. On the following construction, triangle ABC is the original triangle and triangle ECD is the medians triangle. As you see, if the original triangle is equilateral, the triangle of medians is also equilateral.

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2) Medians triangle of an isosceles triangle

Facts A medians triangle of an isosceles triangle is an isosceles triangle. On the following construction, triangle ABC is the original triangle and triangle ECD is the medians triangle. As you see, if the original triangle is isosceles, the triangle of medians is also isosceles.

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3) Medians triangle of a right triangle

Facts A medians triangle of a right triangle is NOT a right triangle. On the following construction, triangle ABC is the original triangle and triangle ECD is the medians triangle. As you see, even though the original triangle is a right triangle(angle ABC is 90 degrees), the triangle of medians is NOT a right triangle(angle CDE is the best candidate, but it is only 85.22 degrees).

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4) Original triangle of a right medians triangle

Facts An original triangle of a right medians triangle is NOT a right triangle. On the following construction, triangle ABC is the original triangle and triangle ECD is the medians triangle. As you see, even though the medians triangle is a right triangle(angle CDE is 90 degrees), the original triangle of a right medians triangle is NOT a right triangle(angleABC is the best candidate, but it is 92.14 degrees).

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5) Conditions when the original triangle and the medians triangle both are right triangles

Facts Conditions when the original triangle and the medians triangle both are right triangles: Triangle ABC and triangle EDC are similar since the corresponding interior angles are congruent. - Angles ABC and EDC, angles BAC and DEC, and angles BCA and DCE are corresponding. - As you see,        m(angle ABC) = m(angle EDC) = 90 degrees        m(angleBAC) = m(angle DEC) = 54.74 degrees        m(angle BCA) = m(angle DCE) = 35.26 degrees - Sides AB and DE, sides BC and DC, and sides CA and EC are corresponding. The ratios of corresponding sides of two trianlges arecongruent. - As you see, all the ratios are 1.15. The corresponding ratios of sides of each triangle are congruent. - As you see,        length of side AB : length of side BC = length of side DE : length of side DC = 0.71        length of side AB : length of side CA = length of side DE : length of side EC = 0.58        length of side BC : length of side CA = length of side DC : length of side EC = 0.82

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