Given that 𝐴𝐵𝐶 is a triangle in which the ratio the measure of angle 𝐴 to the measure of angle 𝐵 to the measure of angle 𝐶 equals four to five to six, determine the triangle’s type in terms of its angles without finding their measures.
First, let’s think about triangle types in terms of their angles. We have acute-angled triangles, right-angled triangles, and obtuse-angled triangles. A right-angled triangle has one angle that measures 90 degrees and an obtuse-angled triangle has one angle that’s greater than 90 degrees. In an acute-angled triangle, all angles measure less than 90 degrees.
90 degrees would be half of the sum of all angles. For an obtuse-angled triangle, one of the angles must be greater than half of the sum of all of the angles. And for a right-angled triangle, one of the angles must be exactly half of the sum of all the angles. So how does this help us here with our ratio?
Our ratio is a ratio of the measures of the three different angles. If we add these three values together, four plus five plus six equals 15. 15 would represent the sum of all the angles. We could make new ratios that represent the part to the whole. Angle 𝐴 is four 15ths of the whole, angle 𝐵 is five 15ths, and angle 𝐶 is six 15ths. The important bit is knowing what half would be. 15 divided by two equals 7.5.
That means if any of the angles are equal to 7.5 out of 15, it would be a right angle. If any of the angles are greater than 7.5 over 15, it would be an obtuse-angled triangle. And if all three of the angles are less than 7.5 out of 15, it would be an acute-angled triangle. Since all three of these angles are less than half of the total, we can say that triangle 𝐴𝐵𝐶 must be an acute-angled triangle.