Video Transcript
Classify the triangle 𝐸𝐹𝐶,
where side length 𝐵𝐹 equals root three centimeters and side length 𝐴𝐸 equals
root six centimeters and where 𝐴𝐵𝐶𝐷 is a rectangle.
To determine the type of
triangle 𝐸𝐹𝐶, we can use the Pythagorean inequality theorem. This tells us that depending on
whether the square of the longest side is greater than, less than, or equal to
the sum of the squares of the other two sides, the angle opposite the longest
side, and therefore the triangle itself, is either obtuse, acute, or right
angled, respectively.
Now in our case, we don’t yet
know the lengths of the sides of our triangle 𝐸𝐹𝐶. But we do know that it’s
inscribed in a rectangle and that the corners of a rectangle are right
angles. So we can use the Pythagorean
theorem for right-angled triangles to find our three missing side lengths. If we start with triangle
𝐹𝐴𝐸, since 𝐴𝐹 is equal to 𝐹𝐵 , that’s root three, we can find the length
of side 𝐸𝐹. 𝐸𝐹 squared equals root six
squared plus root three squared, which is nine. And taking the positive square
root on both sides — positive since we’re looking for the length — we have side
length 𝐸𝐹 equal to three centimeters.
Now if we consider side length
𝐸𝐶, we know that 𝐷𝐸 equals 𝐸𝐴, which is root six centimeters, and that
𝐷𝐶 equals two root three, since it’s the same length as side 𝐴𝐵. So we have 𝐶𝐸 squared equals
𝐸𝐷 squared plus 𝐷𝐶 squared. That’s root six squared plus
two root three squared, which is 18. And taking the positive square
root gives 𝐶𝐸 equal to three root two centimeters.
Using the same method for side
𝐶𝐹, we have 𝐶𝐹 squared equals 𝐶𝐵 squared plus 𝐵𝐹 squared. 𝐶𝐹 squared is therefore
27. And so 𝐶𝐹 is equal to three
root three. We now have all three side
lengths for the triangle 𝐸𝐹𝐶. Now since three root three is
greater than three root two which is greater than three, our longest side is
𝐶𝐹.
And recalling that the angle
with the largest measure in a triangle is always opposite the longest side, we
can apply the Pythagorean inequality theorem to determine the type of the angle
opposite side 𝐶𝐹. That’s angle 𝐶𝐸𝐹. We have that 𝐶𝐹 squared
equals three root three squared, and that’s 27. Next, we have the sum of the
squares of the other two sides, 𝐸𝐹 squared plus 𝐶𝐸 squared, which equals
three squared plus three root two squared. And that’s nine plus 18, which
is also equal to 27. Hence, the square of the
longest side in triangle 𝐸𝐹𝐶 is equal to the sum of the squares of the other
two sides. And so by the third part of the
Pythagorean inequality theorem, angle 𝐶𝐸𝐹 is a right angle.
Now since angle 𝐶𝐸𝐹 is a
right angle and the angles of a triangle sum to 180 degrees, the other two
angles, 𝐸𝐶𝐹 and 𝐸𝐹𝐶, must both be acute angles. Hence, as the angle with the
largest measure in triangle 𝐸𝐹𝐶 is a right angle, triangle 𝐸𝐹𝐶 is a right
triangle.
