In the given problem involving right triangle $ABC$ with $M$ as the midpoint of hypotenuse $AB$, and points $C$ and $D$ such that $DM = CM$, we need to show several properties. Let's address each one systematically:

(i) $\triangle AMC \cong \triangle BMD$

Since $M$ is the midpoint of $AB$, we have:

• $AM = MB$
• $\angle AMC = \angle BMD$ (Each is 90° since $M$ lies on the hypotenuse and $DM = CM$ makes $\triangle MCD$ isosceles with $\angle MCD = \angle MDC$)
• $CM = DM$ (Given)

Thus, by SAS (Side-Angle-Side) congruence postulate, $\triangle AMC \cong \triangle BMD$.

(ii) $\angle DBC$ is a right angle

Since $\triangle AMC \cong \triangle BMD$, corresponding parts of congruent triangles are equal, which implies:

• $\angle DBC = \angle ACB$ (Corresponding angles in congruent triangles)
• $\angle ACB = 90^\circ$ (As $\angle ACB$ is the right angle of the right triangle $ABC$)

Hence, $\angle DBC$ is a right angle.

(iii) $\triangle DBC \cong \triangle ACB$

For triangles $\triangle DBC$ and $\triangle ACB$:

• $\angle DBC = \angle ACB = 90^\circ$ (From previous proofs)
• $BC$ is a common side.
• $BD = AC$ (Since $\triangle AMC \cong \triangle BMD$, $BD = AM$ and $AM = AC$ as $M$ is the midpoint of $AB$)

By RHS (Right-angle Hypotenuse-Side) congruence criterion, $\triangle DBC \cong \triangle ACB$.

(iv) $CM = \frac{1}{2} AB$

Given $M$ is the midpoint of $AB$, by definition, $CM$ (which is $AM$ or $MB$) is exactly half of $AB$.

Would you like more details on any of these proofs or have any questions about the steps taken? Here are some further questions to explore:

1. Why does the midpoint of the hypotenuse of a right triangle always coincide with the circumcenter?
2. How does the property of triangle congruence help in solving real-world problems?
3. Can you derive the Pythagorean theorem from the given triangle and its properties?
4. What are other significant points within a triangle besides the midpoint of the hypotenuse?
5. How can understanding these geometric principles aid in understanding more complex shapes?

Tip: When proving triangles are congruent, always clearly state which criterion you are using (SSS, SAS, ASA, AAS, or RHS) and ensure you have demonstrated each part of that criterion.