Find the length of line segment 𝐶𝐵.
In this question, we’re given a diagram with two different triangles. We’re given the lengths of two sides of each triangle. And we can also see from the markings that this angle, 𝐴𝐶𝐹, is the same size as this angle, 𝐷𝐵𝐹. So, the line segment that we’re asked to work out is the one at the base of the diagram. We’ve got the length of part of it. We know that 𝐶𝐹 is 21 units long and 𝐹𝐵 is two 𝑥 plus eight. While we could just add these two lengths, for example, 21 plus two 𝑥 plus eight would give us two 𝑥 plus 29, but we can assume that what we’re really being asked here is for a numerical value for the length.
So, let’s have a look again at the diagram and see if there’s any way we can work out this length, 𝐹𝐵, perhaps by finding the value of 𝑥. A clue for the message comes from the fact that we’re given this pair of angles which are equal. We might then wonder if these triangles are perhaps similar or congruent. We remember that similar triangles have corresponding angles equal and corresponding sides in proportion. When we’re dealing with congruent triangles, congruent triangles have corresponding angles equal and corresponding sides equal.
However, going by the diagram, these two triangles aren’t the same size, so they’re very likely not to be congruent. So, let’s check if they’re similar. We can note firstly that our pairs of angles are equal. Angle 𝐴𝐶𝐹 is equal to angle 𝐷𝐵𝐹. Next, we could have a look at this angle 𝐴𝐹𝐶, which is marked as a right angle. Using the fact that the angles on a straight line sum to 180 degrees and the line 𝐵𝐶 is a straight line, this means that this angle of 𝐷𝐹𝐵 must also be 90 degrees, since 180 degrees subtract 90 also gives us 90 degrees. Therefore, we note that angle 𝐴𝐹𝐶 is equal to angle 𝐷𝐹𝐵.
This now means that we’ve found two pairs of corresponding angles equal. This fulfills the AA or angle-angle similarity criterion. Now, we’ve proven that triangle 𝐴𝐹𝐶 is similar to triangle 𝐷𝐹𝐵. So, let’s see how this helps us to work out the side length of 𝐹𝐵. As these two triangles are similar, that means remember that corresponding sides are in proportion. If we look at the side 𝐴𝐶, then the side which corresponds to it in triangle 𝐷𝐹𝐵 is this one, 𝐷𝐵. Then, another side on triangle 𝐴𝐹𝐶 that we can look at is the length of 𝐶𝐹. The corresponding side on the other triangle is 𝐹𝐵.
Because the triangles are similar, the sides are in proportion. And as that proportion is equal, then we can write that 𝐴𝐶 over 𝐷𝐵 is equal to 𝐶𝐹 over 𝐹𝐵. We could also have written this statement with the fractions reversed. However, we need to make sure that we keep all the lengths of one triangle either as the numerators or the denominators and make sure we don’t mix them up.
What we do next is simply substitute in the length information that we’re given. This gives us 35 over seven 𝑥 plus six equals 21 over two 𝑥 plus eight. To solve this, we can begin by taking the cross product. So, we have 35 multiplied by two 𝑥 plus eight equals 21 multiplied by seven 𝑥 plus six. In the next step, we could go straight ahead and expand the parentheses on both sides of this equation. However, we might also notice that the values outside the parentheses are both multiples of seven. Dividing through by seven means that we can write it a little more simply. Five multiplied by two 𝑥 plus eight equals three multiplied by seven 𝑥 plus six.
We can now expand the parentheses giving us 10𝑥 plus 40 equals 21𝑥 plus 18. In order to keep a positive value of 𝑥, we can subtract 10𝑥 from both sides. Then, we can subtract 18 from both sides, which leaves us with 22 equals 11𝑥. We can then divide through by 11 which gives us that two equals 𝑥 and 𝑥 is equal to two.
It’s very tempting to stop here and think that we’ve answered the question. But don’t forget we weren’t just asked for 𝑥. We were asked for the length of the line segment 𝐶𝐵. Remember that we said that 𝐶𝐵 is equal to 21 plus this length of two 𝑥 plus eight. We then need to substitute in the value that 𝑥 is equal to two, giving us that 𝐶𝐵 is equal to 21 plus two times two plus eight simplifying to 33. Therefore, we can give the answer that line segment 𝐶𝐵 is equal to 33 length units.