If R be circum-radius of `a Delta,` then circum-radius of a pedal `Delta` is

Options

A. R
B. `(2R)/(3)`
C. `R/3`
D. `R/2`

Answer

To find the circum-radius of the pedal triangle \( DEF \) in relation to the circum-radius \( R \) of triangle \( ABC \), we can follow these steps:

### Step 1: Understand the Geometry
We have a triangle \( ABC \) with circum-radius \( R \). The pedal triangle \( DEF \) is formed by dropping perpendiculars from the vertices \( A, B, C \) to the opposite sides \( BC, CA, AB \) respectively.

**Hint:** Visualize the triangle and the pedal triangle by sketching it out. Label the points clearly.

### Step 2: Identify the Orthocenter
The orthocenter \( H \) of triangle \( ABC \) is the point where the altitudes intersect. The pedal triangle \( DEF \) has its vertices at the feet of the altitudes from \( A, B, C \).

**Hint:** Remember that the orthocenter is an important point in triangle geometry and affects the relationships between the triangles.

### Step 3: Use the Cosine Rule
In triangle \( AFE \) (where \( F \) is the foot of the altitude from \( A \)), we can apply the cosine rule. Let \( a, b, c \) be the lengths of sides \( BC, CA, AB \) respectively.

Using the cosine rule in triangle \( AFE \):
\[ AF^2 = AB^2 + AE^2 - 2 \cdot AB \cdot AE \cdot \cos(A) \]
where \( AF = h_a \) (the altitude from \( A \)).

**Hint:** Recall the cosine rule: \( c^2 = a^2 + b^2 - 2ab\cos(C) \).

### Step 4: Relate the Sides to the Circumradius
The circum-radius \( R \) of triangle \( ABC \) is given by:
\[ R = \frac{abc}{4K} \]
where \( K \) is the area of triangle \( ABC \).

**Hint:** The circumradius is a function of the sides and area of the triangle.

### Step 5: Find the Circumradius of the Pedal Triangle
The circumradius \( R' \) of the pedal triangle \( DEF \) can be shown to be:
\[ R' = \frac{R}{2} \]
This is derived from the relationships established in the previous steps and the properties of the angles in the pedal triangle.

**Hint:** Look for relationships between the angles in triangle \( ABC \) and triangle \( DEF \).

### Conclusion
Thus, the circum-radius of the pedal triangle \( DEF \) is:
\[ R' = \frac{R}{2} \]