In the given rhombus 𝐴𝐵𝐶𝐷, 𝐷𝐵 equals six 𝑥 minus eight and 𝑃𝐵 equals two 𝑥 plus nine. Find the length of line segment 𝑃𝐷.
So here, we have a rhombus. We can recall that a rhombus has four sides of equal length. In our diagram, we can also see the lines 𝐴𝐶 and 𝐷𝐵 are marked. These are the diagonals of the rhombus. We can recall that the diagonals of a rhombus are perpendicular bisectors. So this means that they cross each other at right angles and that each diagonal cuts the other one exactly in half. So to find the length of the line segment 𝑃𝐷, we’ll need to find the length of the diagonal 𝐷𝐵 and halve it. We’re given that 𝐷𝐵 is six 𝑥 minus eight and 𝑃𝐵 equals two 𝑥 plus nine. So let’s see if we can write an equation to represent the relationship here.
We know that 𝑃𝐵 is half a diagonal. So we can write 𝑃𝐵 is equal to half 𝐷𝐵. Substituting our values gives us two 𝑥 plus nine equals half of six 𝑥 minus eight. To simplify the right-hand side then, we simply take half of six 𝑥 and half of negative eight to give us three 𝑥 minus four. For the next stage of rearranging then, we collect our 𝑥 terms on one side. We can keep our highest value on the right-hand side, so we can subtract two 𝑥 from both sides, giving us nine equals 𝑥 minus four since three 𝑥 subtract two 𝑥 is 𝑥.
We then do the inverse operation to subtracting four which is adding four to both sides. So nine plus four is equal to 𝑥, giving us 𝑥 equals 13. We can notice that the question has asked us to find the length of 𝑃𝐷. But we know that 𝑃𝐷 and 𝑃𝐵 will be the same length. Since 𝑃𝐵 was given as two 𝑥 plus nine, we know I need to substitute 𝑥 equals 13 into this equation. 𝑃𝐵 then equals two times 13 plus nine. And since two times 13 is 26, adding nine will give us 35. And therefore, since the two parts of our diagonal are equal, that is, 𝑃𝐷 is equal to 𝑃𝐵, then we can say that 𝑃𝐷 equals 35.