Alicia H. answered • 05/01/19
Creative & Resourceful Math Tutor
You need to use properties of a rhombus here. Here are the two that are important:
- the diagonals in a rhombus (or really in any parallelogram) bisect each other
- the diagonals in a rhombus are perpendicular to one another
You'll need to draw a rhombus with its two diagonals drawn in. If the diagonals bisect each other, then they cut each other in half. That means that you will have 5 cm on one side of the intersection and 5 cm on the other side. We can also label all of the outer side lengths as 13 cm since all four sides are congruent in a rhombus by definition.
Next, if our diagonals are perpendicular to each other, we have to know that perpendicular lines form right angles. This means that the angles inside the rhombus where the diagonals intersect are right angles. Now you have broken up your rhombus into 4 small right triangles.
If you have a right triangle, and you know two of the side lengths and are looking for the third side length, use the Pythagorean Theorem (a2 + b2 = c2). Don't forget to make sure you plug in the hypotenuse (the longest side that is across from the right angle) for c. This means you will plug 13 in for c since 13 cm is the hypotenuse for the right triangle. You then put 5 in for either a or b. It does not matter if you plug the leg length in for a or b, it just has to be one of the legs. Now solve for the missing variable:
52 + b2 = 132
25 + b2 = 169
b2 = 144
b = 12 cm
Since the diagonals of the rhombus bisect each other, and we found one piece of the missing diagonal to be 12 cm, the whole length of the missing diagonal must be 12(2) = 24 cm.
The last piece of the question is to find the area of the entire rhombus, but I would instead find the areas of each of the 4 triangles and then add them for the total area. The area of a triangle is 1⁄2·base·height, so it will be 1/2(5)(12) = 30 cm2. Now multiply 30 by the 4 triangles, and your total area is 120 cm2.
