In the diagram, AD¯¯¯¯¯¯¯¯=13 and DB¯¯¯¯¯¯¯¯=24 . Use the CPCTC Theorem to prove that the diagonal of the rhombus bisects opposite angles by finding the length of DC¯¯¯¯¯¯¯¯ . What is the length of DC¯¯¯¯¯¯¯¯ ?

Answer 1

To prove that the diagonal of a rhombus bisects opposite angles using the CPCTC (Corresponding Parts of Congruent Triangles are Congruent) theorem, we first need to establish the properties of a rhombus.

In any rhombus:

1. All sides are equal in length.
2. The diagonals bisect each other at right angles.
3. The diagonals bisect the opposite angles.

Given that \( AD = 13 \) and \( DB = 24 \), we can find the length of \( DC \).

First, we note that since \( A \) and \( B \) are vertices of the rhombus, we can conclude that \( AD = BC \) and \( AB = CD \). Because \( AB \) and \( CD \) are the sides of the rhombus, we know:

\[ AD = AB = CD = BC \]

However, in this case, since diagonals bisect each other, we know that:

\[ AC = AD + DC \] Since diagonals bisect each other at point \( O \) (the intersection of the diagonals), we have \( AO = \frac{AC}{2} \) and \( BO = \frac{BD}{2} \).

If we denote \( DC \) as \( x \) and since \( AC \) is the sum of \( AD \) and \( DC \), we can set up the equation:

\[ 13 + x \]

Thus, the total length of diagonal \( AC \) is:

\[ AC = AD + DC = 13 + x \]

Also, we know diagonal \( BD \) is \( 24 \), so:

\[ BD = 24 \]

On further note, since these lengths are comparing total lengths of diagonals and not directly impacting finding \( DC \), we recognize that they simply represent the rhombus's properties of diagonals.

To take a clear approach for \( DC \):

We need to establish a triangle from \( A, D, O \), since \( O \) is where the diagonals cross. We know that \( AO + OD = 13 + x \) but \( AO = \frac{AC}{2} \) and \( OD = \frac{BD}{2} = 12\).

Now applying the classic properties of triangles formed in our context and rhombus diagonals, thus the length can be obtained, specifically focusing on sides relating through coordinate placement giving right triangles made by these intersections confirming congruency can be seen:

\[ AO^2 + OD^2 = AD^2\ (AC/2)^2 + (BD/2)^2 = AD^2\ (13+x)^2 + 12^2 = 13^2 \]

The details cannot equate the expectance unless those were defined consistently in other proving aspects, commonly through larger angles relying on specifying crucial triangle distinctivity.

Therefore the full assignable extent \( DC \) can be determined heuristically or via isolating conjunctive geometric definitions.

For lengths finally producing,

Thus we find:

\[ DC = 13 \]

This part interprets that within the rhombus properties, consistent angular measure allows confirming that each diagonal travels through equal properties thus maintaining \( AO = OC \) yielding decisive length angles across clarified definitions of a rhombus geometry.

This reflects on showing that the diagonals themselves confirm exactly \( x = DC; \) remaining part \( OA = (24/2) = 12 \) thus reflecting proportions through their intersection clarifying rotational dimensions signify:

So finally:

\[ \boxed{13} \]

is the length of \( DC \).