Question

If PQRS is a convex quadrilateral with 3, 4, 5 and 6 points marked on sides PQ, QR, RS and PS respectively. Then, the number of triangles with vertices on different sides is

• A. 220
• B. 270
• C. 282
• D. 342

Hint:

Choose sides first, then multiply the number of points on each selected side.

Answer 1

The Correct Option is D

To solve this problem, we need to find the number of triangles that can be formed using vertices on different sides of the convex quadrilateral PQRS.

Let's analyze the number of points on each side:

• PQ has 3 points.
• QR has 4 points.
• RS has 5 points.
• PS has 6 points.

We need to select one point from each of three different sides to form a triangle. So, the number of possible triangles can be calculated by selecting one side for each vertex of the triangle.

We can select the sides in the following combinations: (PQ, QR, RS), (PQ, QR, PS), (PQ, RS, PS), and (QR, RS, PS).

Now, let's calculate the number of triangles from each configuration:

1. For (PQ, QR, RS): Select 1 point from PQ \( \times \) 1 point from QR \( \times \) 1 point from RS.
2. For (PQ, QR, PS): Select 1 point from PQ \( \times \) 1 point from QR \( \times \) 1 point from PS.
3. For (PQ, RS, PS): Select 1 point from PQ \( \times \) 1 point from RS \( \times \) 1 point from PS.
4. For (QR, RS, PS): Select 1 point from QR \( \times \) 1 point from RS \( \times \) 1 point from PS.

Adding all these, the total number of triangles is:

\(60 + 72 + 90 + 120 = 342\)

Therefore, the number of triangles with vertices on different sides of the quadrilateral is 342.