Video Transcript
If 𝐴𝐵𝐶 is a triangle whose side
lengths are 11 centimeters, 26.4 centimeters, and 28.6 centimeters and is similar to
a triangle 𝑋𝑌𝑍, determine the type of triangle 𝑋𝑌𝑍 in terms of its angles.
In this question, we are given the
side lengths of a triangle 𝐴𝐵𝐶 and asked to determine the type of a similar
triangle 𝑋𝑌𝑍. We can start by recalling that
similar triangles have all of their corresponding angles congruent. So the types of triangle 𝐴𝐵𝐶 and
𝑋𝑌𝑍 will be the same. Therefore, we can determine the
type of triangle 𝐴𝐵𝐶 instead. We can do this by applying the
Pythagorean inequality theorem to the given lengths.
We can recall that this tells us we
can determine the type of a triangle from its side lengths by comparing the square
of the length of the longest side with the sum of the squares of the two shorter
sides in the triangle. In general, if 𝑃𝑄 is the longest
side in triangle 𝑃𝑄𝑅, then we can compare the values of 𝑃𝑄 squared and 𝑃𝑅
squared plus 𝑄𝑅 squared to check the measure of the largest angle, which is at
𝑅.
In our case, we have that the
longest side in triangle 𝐴𝐵𝐶 has length 28.6 centimeters. So, we can square this to find that
28.6 squared is equal to 817.96. We want to compare this to the sum
of the squares of the lengths of the two shorter sides. We calculate that 11 squared plus
26.4 squared is equal to 817.96. We see that the side lengths in
triangle 𝐴𝐵𝐶 satisfy the equation in the Pythagorean theorem. And we know that for this to be
true, triangle 𝐴𝐵𝐶 must be a right triangle.
Finally, since triangle 𝑋𝑌𝑍 is
similar to triangle 𝐴𝐵𝐶, the corresponding angles are congruent. So triangle 𝑋𝑌𝑍 must also be a
right triangle.
