In right triangle ABC with right angle at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D, such that DM = CM. Point D is joined to B. If CD = 10 cm, and BD = 6 cm, find the value of CM.

Correct Answer: 5 cm

Step-by-step Solution:

1. Understand the Given Information:

We are given a right-angled triangle ABC with the right angle at vertex C ($\angle ACB={90}^{\circ }$).

M is the midpoint of the hypotenuse AB, which means $AM=BM$.

Line segment CM is produced to point D such that $DM=CM$, making M the midpoint of segment CD as well.

We are given that the length of CD is 10 cm and BD is 6 cm.

2. Relationship between CM and CD:

Since point C is joined to M and extended to D such that $DM=CM$, the total length of segment CD is:

$CD=CM+DM$

Since $DM=CM$, we can substitute $DM$ with $CM$:

$CD=CM+CM=2\cdot CM$

3. Calculate the value of CM:

We are given $CD=10\text{cm}$.

Using the relationship derived above:

$2\cdot CM=10\text{cm}$

$CM=\frac{10}{2}\text{cm}=5\text{cm}$

Thus, the value of CM is 5 cm.