Determine whether triangle 𝐵𝐶𝐷 is an obtuse or acute triangle.
Looking at this figure, we’ve been given the three side lengths of the larger triangle 𝐴𝐵𝐶, and we know that the line 𝐵𝐷 is a median of the larger triangle. At this point, visually, it looks like the larger triangle 𝐴𝐵𝐶 is a right triangle. But because we haven’t been given this information, we need to confirm that that is true, which we can do using the Pythagorean theorem. If this is, in fact, a right angle, 108 squared plus 81 squared should equal 135 squared. When we calculate all these values, we find that that is a true statement. And we can say that the angle at 𝐵 is a right angle.
But we’re trying to identify what type of triangle 𝐵𝐶𝐷 is. We know one of its side lengths, and we know that its other side length will be half of 135 because the median 𝐵𝐷 divides 135 into two equal portions. Both of these equal portions will be 67.5 centimeters in length. At this point, it might seem like there’s very little we can do, but we need to remember something about the median of a right triangle. And that is that in a right triangle, the median is equal to half of the hypotenuse.
Our hypotenuse was 135. And that means the line segment 𝐵𝐷, the median, is also equal to 67.5 centimeters. That means we’re saying triangle 𝐵𝐶𝐷 is an isosceles triangle since line segments 𝐶𝐷 and 𝐵𝐷 both measure 67.5 centimeters and the third side, the longest side, measures 81 centimeters. But how do we definitively say if triangle 𝐵𝐶𝐷 is an obtuse or an acute triangle?
We know that these two angles will be equal to each other as they’re opposite the equal side lengths. We can also say that both of these angles will be less than 90 degrees. And that means to determine if this is an obtuse or an acute triangle, we need to think about the angle opposite the line segment 81 centimeters. To do this, we can use a kind of modified Pythagorean theorem statement. In an obtuse triangle, 𝑐 squared, the longest side squared, will be greater than the sum of the two smaller sides squared. Similarly, in an acute triangle, the square of the longest side will be less than the sum of the squares of the two smaller sides.
And that means we need to plug in our longest side for 𝑐 and the two smallest sides for 𝑎 and 𝑏. How does 81 squared compare to 67.5 squared plus 67.5 squared? When we do those calculations, we see we’re comparing 6561 to 9112.5. 6561 is less than 9112.5.
Since the square of the longest side was less than the sum of the squares of the two smaller sides, this fits the acute triangle. And so, we can say that triangle 𝐵𝐶𝐷 is an acute triangle.