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Median of a right isosceles triangleTo show that the median of a right isosceles triangle is half the hypotenuse, start with a right triangle and then draw the median (shown in red)  This will also mean that angle BDA = 90 degrees. Therefore, we can use the Pythagorean Theorem for triangle ABC and triangle and triangle ABD Let us show that triangle ABD and triangle ADC are congruent by SSS (three equal sides) We already know that segment AB = segment AC since triangle ABC is isosceles Since segment AD is the median of segment BC, segment BD = segment DC Finally, segment AD is a common side, so it is equal for both triangles We have found 3 equal sides, so triangle ABD and triangle ADC are congruent and we can do the aforementioned Now, let's use the Pythagorean Theorem for triangle ABC and triangle and triangle ABD. The focus now will be on lengths of sides  x2 = y2 + y2 x2 = 2y2 y2 = (x2) / 2 For triangle ABD, y2 = z2 + (x/2)2 Substitute (x2) / 2 for y2 We get: (x2) / 2 = z2 + (x/2)2 (x2) / 2 = z2 + (x2) / 4 (x2) / 2 - (x2) / 4 = z2 (2x2) / 4 - (x2) / 4 = z2 (x2) / 4 = z2 x/2 = z (Done!) Learn much more than a right isosceles triangle. Buy a comprehensive geometric formulas ebook. All geometric formulas are explained with well selected word problems  Need a Quick Answer to your Basic Mathematics Problems? Get an answer in 10 minutes or less from a math expert! Justanswer features top-notch math experts handpicked by personnel after they have taken and passed a rigourous math test and after their credentials have been verified by a third party
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