In which one of the following triangles does the orthocentre lie in the exterior of the triangle.

Options

A. `Delta ABC`, where in, `angle A = angle B = angle C = 60^@`
B. `Delta PQR`, wherein, `angle P = 40^(@) , angle Q = 30^(@) , angle R = 110^(@)`
C. `Delta XYZ`, wherein, `angle X = 80^(@), angle Y = 60^(@), angle Z = 40^(@)`
D. `Delta DEF`, wherein , `angle D= 52^(@), angle E = 90^(@), angle F = 38^(@)`

Answer

To determine in which triangle the orthocenter lies outside the triangle, we need to understand the properties of the orthocenter in relation to the angles of the triangle.

### Step-by-Step Solution:

1. **Understanding the Orthocenter**: The orthocenter of a triangle is the point where the three altitudes intersect. The position of the orthocenter depends on the type of triangle:
- In an acute triangle, the orthocenter lies inside the triangle.
- In a right triangle, the orthocenter lies at the vertex of the right angle.
- In an obtuse triangle, the orthocenter lies outside the triangle.

2. **Identifying the Type of Triangle**: To find out in which triangle the orthocenter is outside, we need to look for an obtuse triangle. An obtuse triangle has one angle greater than 90 degrees.

3. **Analyzing the Given Angles**: The problem mentions angles:
- Angle P = 40 degrees
- Angle Q = 30 degrees
- Therefore, Angle R = 180 - (Angle P + Angle Q) = 180 - (40 + 30) = 110 degrees.

4. **Determining the Type of Triangle**: Since Angle R is 110 degrees, which is greater than 90 degrees, triangle PQR is an obtuse triangle.

5. **Conclusion**: Since triangle PQR has an obtuse angle, the orthocenter of triangle PQR lies outside the triangle.

### Final Answer:
The orthocenter lies outside the triangle PQR.