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CCSS.Math: ,

so we have this rectangle right over here and we're told that the length of a B is equal to 1 so that's labeled right over there a B is equal to 1 and then they tell us that B E and B D trisect angle ABC so b e and B D trisect angle ABC so trisect means dividing it into three three equal angles so that means that this angle is equal to this angle is equal to that angle and what they want us to figure out is what is the perimeter of triangle B edie triangle B II D so it's kind of this middle triangle in the rectangle right over here so at first it seems like a pretty hard problem because you're like well what is the width of this rectangle how can I even start on this they've only Givens one side here they've actually given us a lot of information given that we do know that this is a rectangle we have four sides and then we have four angles and they're the sides are all parallel to each other and that the angles are all 90 degrees which is more than enough information to know that this is definitely a rectangle and so one thing we do know is that opposite sides of a rectangle are the same length so if this side is one then this side right over there is also one the other thing we know is that this angle is trisected now we know what the measure of this angle is it was a right angle it was a 90 degree angle so if it's divided into three equal parts that tells us that this angle right over here is 30 degrees this angle right over here is 30 degrees and then this angle right over here is 30 degrees and then we see that we're dealing with a couple of 30-60-90 triangles this one is 30 90 so this other side right over here needs to be 60 degrees so that side right over there needs to be 60 degrees this triangle right over here you have 30 you have 90 so this one has to be 60 degrees they have to add up to 180 30 60 90 triangle 30 60 90 triangle and you can also figure out the measures of this triangle although it's not going to be a right triangle but knowing what we know about 30 60 90 triangles if we just have one side of them we can actually figure out the other sides so for example here we have the shortest side we have the side opposite the 30-degree side now whatever if the 30 every side is 1 then the 60-degree side is going to be square root of 3 times that so this length right over here is going to be square root of 3 and that's pretty useful because we now just figure it out the length of the entire base of this rectangle right over there and we just use our knowledge of 30-60-90 triangles if you see if that was a little bit mysterious how I came up with that I encourage you to watch that video we know that 30-60-90 triangles so we know that 30-60-90 triangles their sides are in the ratio of 1 to square root of 3 to 2 so if this is 1 this is a 30-degree side this is going to be square root of 3 times that and then the hypotenuse right over here is going to be 2 times that so this length right over here is going to be 2 times the side right over here so 2 times 1 is just 2 so that's pretty interesting let's see if we can do something similar with this side right over here here the 1 is not the side opposite the 30-degree side here the 1 is the side opposite the 60-degree side this is the one opposite the 60-degree side so once again if we multiply this side times square root of 3 we should get this side right over here this is the 60 remember this one so right over here this one this is the 60-degree side so this is going to be this has to be 1 square roots of threes of this side let me write this down 1 over the square root of 3 and the whole reason I wait I was able to get this as well whatever this side if I multiply it by the square root of 3 I should get this side right over here I should get the 60-degree side the side opposite the 60-degree angle or if I take the 60-degree side if I divide it by the square root of 3 I should get the shortest side the 30-degree side so if I start with the 60-degree side divided by the square root of 3 I get that right over there and then the hypotenuse is always going to be twice the sahte twice the length of the side opposite the 30-degree angle so this is the side opposite the 30-degree angle the hypotenuse is always twice that so this is the side opposite the 30-degree angle the hypotenuse is going to be twice that it is going to be 2 over the square root of 3 so we're doing pretty good we have to figure out the perimeter of this inner triangle right over here we already figured out one length is 2 we figure out another length is 2 square roots of 3 and then we all we have to really figure out is what Edie is and we can do that because we know that ad is going to be the same thing as BC we know that this entire length because we're dealing with a rectangle is the square root of three this entire length if that entire length of square root of three if this part this a E is 1 over the square root of 3 then this length right over here edie is going to be square root of 3 minus 1 over the square root of 3 that length minus that length right over there and now to find the perimeter is pretty straightforward we just have to add these things up and simplify it so it's going to be 2 so let me write this perimeter of triangle be e d is equal to this is short for perimeter I just didn't feel like writing the whole word is equal to 2 over the square root of 3 plus square root of 3 minus 1 over the square root of 3 minus 1 over the square root of 3 plus 2 and now this just boils down to simplifying radicals you can take a calculator out and get some type of decimal approximation for it let's see if we have 2 square roots of 3 minus 1 square root of 3 that'll leave us with 1 1 over the square root of 3/2 divided by 3/2 over square root of 3 minus 1 over the square root 3 is 1 over the square root of 3 and then you have plus the square root of 3 plus 2 and let's see I can rationalize this if I multiply the numerator in the denominator by the square root of 3 this gives me the square root of 3 over 3 plus the square root of 3 which is I can rewrite that as plus 3 square roots of 3 over 3 right I just multiply this times 3 over 3 plus plus 2 and so this gives us this is the drumroll part now so 1 square root of 3 plus 3 square roots of 3 and all of that over 3 gives us 4 square roots of 3 over 3 plus 2 or you could put the 2 first some people like to write the the non irrational part before the irrational part but we're done we figured out the perimeter we figured out the perimeter of this inner triangle B edie right there you