Given that triangles 𝐴𝐵𝐶 and 𝐴 prime 𝐵 prime 𝐶 prime are similar, work out the value of 𝑥.
So the key factor here is that these two triangles are similar, which means that they have proportional side lengths. The ratio between corresponding pairs of sides is the same. So the ratio between 𝐵𝐶 and 𝐵 prime 𝐶 prime is the same as the ratio between 𝐴𝐶 and 𝐴 prime 𝐶 prime.
So we have that 𝐵𝐶 over 𝐵 prime 𝐶 prime is equal to 𝐴𝐶 over 𝐴 prime 𝐶 prime. We’ve been given the lengths of all of these sides exactly for the smaller triangle and in terms of the variable 𝑥 for the larger triangle. Let’s substitute in the expressions or the values for each of the sides.
For the green sides, the hypotenuse of the triangles, we have two 𝑥 plus one over six. And for the pink side, we have 𝑥 plus three over four. Now what we’ve done is form an equation which we now want to solve in order to work out the value of 𝑥. From this stage here, the question is purely an algebraic one.
Now we have a four and a six in the denominator of these fractions. So we’ll cross multiply. This gives four lots of two 𝑥 plus one is equal to six lots of 𝑥 plus three. Next, we need to expand the brackets on each side of the equation. So we have eight 𝑥 plus four is equal to six 𝑥 plus 18.
Now there are 𝑥s on both sides of this equation. So in order to have 𝑥s only on one side, in this case the left, we need to subtract six 𝑥 from both sides. This leads us to two 𝑥 plus four is equal to 18. Next, we need to subtract four from both sides of the equation.
And we now have that two 𝑥 is equal to 14. The final step is to divide both sides of the equation by two. So this gives us the solution to the equation and our answer to the problem: 𝑥 is equal to seven.