Find the value of 𝑧 in the following parallelogram.
The diagram shows a quadrilateral, which we’re told is a parallelogram. Its diagonals have also been drawn in. The diagonals intersect at a point in the interior of the parallelogram. The two parts of one of the diagonals have been given as expressions in terms of the variable 𝑧, whose value we’re looking to calculate.
In order to answer this question, we need to think about what we know about the diagonals of parallelograms. An important key fact is this the diagonals of a parallelogram bisect each other. What does this mean well? The word “bisect” means to cut in half. So the two portions of each diagonal are the same length. So for the diagonal whose lengths have been given in terms of the variable 𝑧, this means that the two expressions must be equal to each other.
We can express this as an equation six 𝑧 minus five is equal to 𝑧 plus 10. The problem has now become an algebraic one. In order to find the value of 𝑧, we need to solve this equation. First I’m going to add five to both sides of the equation. This gives six 𝑧 is equal to 𝑧 plus 15.
Now there are terms involving 𝑧 on both sides of the equation. So next, I’d like to subtract 𝑧 from both sides. Subtracting 𝑧 from the left-hand side gives five 𝑧. And subtracting 𝑧 from the right-hand side gives 15.
So now we have five 𝑧 is equal to 15. The final step in solving this equation is to divide both sides by five. This gives the solution to the problem: the value of 𝑧 is three.