In a rhombus, the diagonals bisect each other at right angles and also bisect the opposite angles. Given the segments AD and DB, we find the other diagonal (which we can denote as AC) since we can use these lengths to find the lengths of triangles formed by the diagonals.
First, let's denote AC as the diagonal that we want to analyze. In the rhombus, the diagonals bisect each other. Therefore, if AD and DB are segments we're given (with AD = 13 and DB = 24), we can find the midpoint of diagonal AC where both diagonals intersect, which is also the center point of the rhombus.
Using the Pythagorean theorem on triangle ADB, we first find the length of AB (which is also equal to the lengths of all the sides of the rhombus because all sides are equal).
Since:
- AD = 13
- DB = 24
Thus, the entire diagonal AC = AD + DB = 13 + 24 = 37.
Now, we need to find the length DC. Since D is the midpoint of the diagonal AC, we can compute the half-lengths:
- DC = \(\frac{1}{2} AC = \frac{1}{2} (37) = 18.5\)
However, since this length isn't provided, it may seem we are not on the right track.
To solve the problem correctly utilizing properties of the rhombus, we recognize that triangles ADB and ADC are congruent. With the lengths given, splitting them equally implies that:
- \(DB (24)\) represents that if D is at the midpoint, then:
- \(AD (13)\) fully represents opposite angles DAB and ABC.
Thus, we determine:
- DC = DB as opposite angles get bisected. We can check potential answers provided.
Hence, given options: 5, 24, 12, 13 and evaluating they are lengths divided by the bisection, the only feasible option is responding based on congruency:
DC = 13 would be correct based on equality of bisected parts along diagonals via rhombus properties.
Thus the answer is: 13.
1 answer