Video Transcript
𝐴𝐵𝐶𝐷 is a
parallelogram. If 𝐴𝐶 equals 13 centimeters,
𝐴𝐷 equals 13 centimeters, and 𝐷𝐶 equals five centimeters, what is the type
of triangle 𝐴𝐷𝐶?
We see that triangle 𝐴𝐷𝐶 is
an isosceles triangle. And recalling that the angle in
a triangle with the greatest measure is opposite the longest side, in triangle
𝐴𝐷𝐶 the angles at 𝐶 and 𝐷, which are equal, will have the largest
measure. Choosing either one of the
angles at 𝐶 and 𝐷, we can use the Pythagorean inequality theorem to confirm
that these angles are acute.
Taking the angle at 𝐷 to work
on, this theorem tells us three things. First, that if the square of
the longest side is greater than the sum of the squares of the other two sides,
then the angle opposite the longest side is an obtuse angle. Second, if the square of the
longest side is less than the sum of squares of the other two sides, the angle
is acute. And third, if the square of the
longest side is equal to the sum of the squares of the other two, then the angle
opposite is a right angle.
In our case, we have 𝐴𝐶
squared, that is 13 squared, equals 169 and that 𝐴𝐷 squared plus 𝐷𝐶 squared
equals 13 squared plus five squared. And that’s equal to 194. Hence, 𝐴𝐶 squared is less
than 𝐴𝐷 squared plus 𝐷𝐶 squared. And so angle 𝐶𝐷𝐴 is an acute
angle. Angle 𝐴𝐶𝐷 is the same, so
this is also acute. And since these angles have the
largest measure in triangle 𝐴𝐷𝐶, angle 𝐶𝐴𝐷 must be smaller than them. Hence, the third angle, angle
𝐶𝐴𝐷 is also acute. Since all three angles are
acute and, in particular, the angle with the largest measure is acute, triangle
𝐴𝐷𝐶 is an acute triangle.
