𝐴𝐵𝐶𝐷 is a rhombus. If the measure of angle 𝐷𝑃𝐶 equals two 𝑥 plus 22 degrees, find the value of 𝑥.
The first thing to do here is to recall what it means to be a rhombus. A rhombus is defined as a quadrilateral that has all four sides equal. We could show this on the diagram with the mark on the line to indicate that these four lengths are of the same size. We’re given in the question that the angle 𝐷𝑃𝐶 measures two 𝑥 plus 22 degrees. And we’re asked to find the value of 𝑥. We might realize at this point that simply knowing that the four lengths are the same and being given the value of one angle won’t help us unless we recall another fact about rhombuses.
In a rhombus, the diagonals are perpendicular bisectors. The line 𝐴𝐶 is a diagonal, and so is the line 𝐵𝐷. Because these are bisectors, we could say that 𝐷𝑃 is equal to 𝑃𝐵 and 𝐴𝑃 is equal to 𝑃𝐶. The fact that they’re perpendicular means that the four angles created at this point 𝑃 will all be 90 degrees. We could therefore set up the equation that this angle 𝐷𝑃𝐶, which is two 𝑥 plus 22 degrees, must also be equal to 90 degrees. We could then solve it to find the value of 𝑥.
We would begin by subtracting 22 degrees from both sides of the equation, which gives us that two 𝑥 equals 68 degrees. Dividing both sides by two, we would find that 𝑥 equals 34. We don’t necessarily need to include the degrees symbol as it was already included as part of the angle measurement of 𝐷𝑃𝐶. So we can give our answer 𝑥 equals 34.