Determine the lengths of 𝐵𝐷 and 𝐴𝐵.
To start this problem, we’re actually going to look to find 𝐵𝐷 first. And to enable us to do that, we’re gonna have a look at a little relationship that applies to this type of triangle. And the relationship we’re looking at is actually due to this triangle being a right triangle. And it is that, as I’ve pointed out with the pink arrow, because we actually have a right angle within.
So the relationship is that in a right triangle the length of the median from the vertex of the right angle is equal to half the length of the hypotenuse. And what this means in practice is actually that our length 𝐵𝐷 that we’re trying to find is actually equal to half of 𝐴𝐶. And that’s because 𝐴𝐶 is our hypotenuse. And we know that it’s our hypotenuse because it’s actually opposite the right angle. And it’s also the longest side.
Okay, great! So let’s use this information to help us find 𝐵𝐷. So therefore, we know that 𝐵𝐷 is gonna be equal to a half multiplied by 49. And that’s 49 because as we said that’s the length of 𝐴𝐶. So therefore, 𝐵𝐷 is equal to 24.5. We’ve actually solved the first part of the question. So now let’s move on and find the length of 𝐴𝐵. Well, the relationship that we’re going to use to help us find the length of 𝐴𝐵 is this one.
For any 30-degree, 60-degree, 90-degree triangle, the side opposite the 30-degree angle is half the hypotenuse. And we know that our triangle is a 30,60,90-triangle because we’ve got a right angle, which we’ve pointed out earlier. So that’s 90 degrees. We’ve got 30 degrees marked onto our triangle. And then, therefore, the other angle must be 60 degrees because the angles in a triangle are 180. 180 minus 90 minus 30 gives us 60 degrees.
Okay, great! Let’s use this relationship to actually find 𝐴𝐵. But what does it mean in practice? Well it means that 𝐴𝐵 is going to be equal to a half of 𝐴𝐶. So we’re gonna get 𝐴𝐵 is equal to a half multiplied by 49. So therefore, we get that 𝐴𝐵 is equal to 24.5. So now we’ve found the length of 𝐵𝐷 and 𝐴𝐵. We just need to add in the units. So our units are centimeters. So therefore, we can say the lengths of 𝐵𝐷 and 𝐴𝐵 are both 24.5 centimeters.