in this video we're going to focus on finding the missing side of a triangle if we're given two similar triangles so let's start with an example so let's say that these two triangles are similar to each other the triangle on the left let's call it triangle ABC and this is going to be triangle DF and let's say that segment AB is 8 and BC is 6 de is 12 units long and our goal is to find the Miss inside EF so we're going to place an X there so how can we find the value of x in addition let's say that angle a is similar to angle D and angle C is congruent to angle f so with this information what can we do to find the value of x what we need to do is set up a proportion let's call this triangle T1 and this triangle T2 so I'm going to have two fractions separated by an equal sign on the left I'm going to put the information associated with triangle one on the right triangle two on the numerator of triangle one that's going to be 8 now 8 is similar to 12 these two sides correspond to each other this one six is going to be the denominator for triangle one and that is similar to X for triangle 2 so that's how you can set up the proportion now in order to solve it we need to cross multiply so 8 * X that's 8 x and that's going to be equal to 6 * 12 which is 72 so now all we need to do is divide both sides by 8 72 / 8 is 9 so that's going to be the answer X is equal to 9 now let's work on another example so let's say this is triangle ABC and this is d EF let's say that AB is 18 and BC is 24 and let's say that angle a corresponds to angle F angle B corresponds to angle e and C corresponds to angle d and let's say this is y now let's say uh de is 16 actually not 16 let's make that 30 and let's say that EF is X and let's make DF 40 and go ahead and find the values of X and Y in this problem so let's start with X this is going to be for triangle one and triangle two now we're going to say triangle one is the one on the left angle a which is associated with 24 that corresponds to angle F so if we put 24 on top of triangle one we need to put 30 on top of triangle 2 those two sides correspond to each other now 18 which is associated with angle C it's opposite to it is similar to angle D which is associated with X so we're going to put 18 and X X so now let's cross multiply so first we have 18 * 30 and that's equal to 24 * X now to make the math easier you can break down the numbers into smaller numbers for example 30 is 6 * 5 and 18 is 9 * 2 24 is 6 * 4 and 4 is 2 * 2 so notice that you can cancel a six and you can cancel a two which will make life easier 9 * 5 is 45 so 45 is equal to 2X and if you divide both sides by two x is 45 over 2 which as a decimal is 22.5 so that's the value of x now let's find the value of y let's set up another proportion between triangles 1 and two so just like in the last example we know that angle a is still congruent to angle F so 24 corresponds to 30 so we can put that on top again now angle y I mean not angle y but Side Y has to correspond to 40 angle B and angle e are congruent to each other so Y and 40 they're on similar sides so now that we have the uh proportion we can go ahead and solve it so let's cross multiply we're going to have 30 * Y is equal to 24 * 40 so let's see if we can simplify our answer without the use of a calculator 30 is 3 * 10 24 is 3 * 8 and 40 is 4 * 10 so we can cancel a 10 from both sides and we can cancel a three so what we have left over is y is equal to 8 * 4 and 8 * 4 is 32 so that's the value of y in this problem here's another example that you could work on for the sake of practice so once again this is going to be triangle ABC and this is triangle DF let's say that this is 15 and this is 9 and this is X and this side is x + 10 and let's say that angle a corresponds to angle D angle C corresponds to angle F and angle B corresponds to angle e so with this information go ahead and find the value of x so just like before let's create a proportion 15 corresponds to x + 10 because they're across two angles that are congruent angle C is congruent to angle F now 9 corresponds to X because angle B and angle e are congruent to each other so now that we have this proportion we can go ahead and find the value of x so let's cross multiply 15 * X is simply 15x x 9 * x + 10 we need to use the distributive property so that's going to be 9x + 90 9 * 10 is 90 so now let's subtract both sides by 9x 15x - 9x is 6X so 6X is equal to 90 all we need to do is divide by 6 90 ID 6 is 15 so that's the the value of x so segment DF is 15 units long D is 15 + 10 or 25 so that's how you can find the two missing sides of this triangle let's try one last example so let's call this angle a b c d and e let's say that AB is 8 AC is is 5 BC is 7 DC is X De is 12 and angle a is congruent to angle e and angle B is congruent to angle D angle ACB is congruent to dce so with this information go ahead and find X let's call the top triangle triangle one and the bottom one triangle 2 so let's set a proportion between T1 and T2 now angle a is congruent to angle e so that means s and X are similar to each other angle C well these two angles are equal to each other so 8 and 12 correspond to each other so now let's cross multiply 8X is equal to 7 * 12 8 is 4 * 2 12 is 4 * 3 so we can cancel a four therefore 2X X is equal to 7 * 3 which is 21 so X is 21 / 2 which as a decimal that's 10.5 so that's the value of x in this problem so now you know how to solve similar triangles using proportions for