**Core Concept**
The question is testing the clinical diagnosis of a congenital hip anomaly, specifically a developmental dysplasia of the hip (DDH). DDH is a condition where the hip joint does not form properly, leading to instability and potential dislocation of the femoral head.

**Why the Correct Answer is Right**
The child's symptoms and examination findings suggest a right hip disorder. The Trendelenburg sign is positive, indicating weakness of the gluteus medius muscle, which is responsible for stabilizing the hip during abduction. The absence of telescoping suggests that the femoral head is not dislocated, but rather subluxated or partially dislocated. This pattern is consistent with a dislocatable or subluxable hip, where the femoral head can be partially displaced from the acetabulum.

**Why Each Wrong Option is Incorrect**

* **Option A:** Coxa vara is a condition where the angle between the femoral neck and shaft is decreased, leading to a short limb. However, coxa vara does not typically cause limited abduction or a positive Trendelenburg sign.
* **Option B:** Perthes disease is a condition where the femoral head undergoes necrosis, leading to a painful limp. However, Perthes disease typically presents with pain and limited mobility, not a short limb or positive Trendelenburg sign.
* **Option D:** Osteomyelitis is an infection of the bone, which can cause pain and limp. However, osteomyelitis does not typically cause a short limb or positive Trendelenburg sign.

**Clinical Pearl / High-Yield Fact**
In children with DDH, the affected limb is typically shorter than the unaffected limb, and the Trendelenburg sign is often positive. However, the absence of telescoping (also known as the Ortolani sign) suggests that the femoral head is not dislocated, but rather subluxated or partially dislocated.

**Correct Answer: C. Developmental Dysplasia of the Hip (DDH)**