In triangle 𝑋𝑌𝑍, 𝑌𝑍 squared is greater than 𝑋𝑍 squared minus 𝑋𝑌 squared. What type of angle is 𝑌?
If we consider any triangle 𝑋𝑌𝑍, there are three possibilities for angle 𝑌. The first diagram shows that angle 𝑌 is a right angle. The second diagram shows angle 𝑌 is an acute angle, as it is less than 90 degrees or a right angle. In the third diagram, angle 𝑌 is an obtuse angle, as it is greater than 90 degrees but less than 180 degrees. We need to consider the relationship between the lengths in the triangle, 𝑌𝑍, 𝑋𝑍, and 𝑋𝑌. In this question, we need to decide which triangle corresponds to the inequality 𝑌𝑍 squared is greater than 𝑋𝑍 squared minus 𝑋𝑌 squared.
Pythagoras’ theorem states that in any right-angled triangle, 𝑎 squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the length of the longest side known as the hypotenuse. This means that in our first diagram, 𝑋𝑌 squared plus 𝑌𝑍 squared is equal to 𝑋𝑍 squared. As angle 𝑌 gets smaller, the length 𝑋𝑍 also gets smaller. This means that in the second diagram, the sum of the other two sides squared will be greater than 𝑋𝑍 squared. Conversely, as angle 𝑌 gets larger, the length 𝑋𝑍 gets larger. This means that in our third diagram, 𝑋𝑌 squared plus 𝑌𝑍 squared is less than 𝑋𝑍 squared.
We can now work out which of our three triangles matches the initial inequality. We can rule out the answer right angle as this has an equation and not an inequality. Let’s consider the initial inequality 𝑌𝑍 squared is greater than 𝑋𝑍 squared minus 𝑋𝑌 squared. Adding 𝑋𝑌 squared to both sides of this inequality gives us 𝑋𝑌 squared plus 𝑌𝑍 squared is greater than 𝑋𝑍 squared. This corresponds to the second diagram. We can therefore conclude that, in this case, angle 𝑌 is acute. It is less than 90 degrees.