ABCD is a rectangle length of whose two consecutive sides are respectively 9 cm and 40 cm. E and F are respectively midpoints of sides AB and CD. A is joined of E and F is joined to C. They respectively intersect BD at P and Q. Length of PQ is-

Options

A. A)`31/(3)"cm"`
B. B)`41/(3)"cm"`
C. C)`49/(3)"cm"`
D. D)`47/(3)"cm"`

Answer

To solve the problem step by step, we will follow the geometric properties of a rectangle and the midpoints of its sides.

### Step-by-Step Solution:

1. **Identify the Rectangle and Midpoints**:
- Let rectangle ABCD have sides AB = 9 cm and BC = 40 cm.
- The midpoints E and F are defined as follows:
- E is the midpoint of AB.
- F is the midpoint of CD.

2. **Coordinates of Points**:
- Assign coordinates to the vertices of the rectangle:
- A(0, 0)
- B(0, 9)
- C(40, 9)
- D(40, 0)
- Calculate the midpoints:
- E = ((0 + 0)/2, (0 + 9)/2) = (0, 4.5)
- F = ((40 + 40)/2, (9 + 0)/2) = (40, 4.5)

3. **Equation of Diagonal BD**:
- The coordinates of B and D are B(0, 9) and D(40, 0).
- The slope of line BD = (0 - 9) / (40 - 0) = -9/40.
- The equation of line BD using point-slope form (y - y1 = m(x - x1)):
- y - 9 = -9/40(x - 0)
- y = -9/40x + 9

4. **Equation of Line EF**:
- The coordinates of E and F are E(0, 4.5) and F(40, 4.5).
- Since E and F have the same y-coordinate, line EF is horizontal:
- y = 4.5

5. **Finding Intersection Points P and Q**:
- To find point P (intersection of EF and BD):
- Set y = 4.5 in the equation of BD:
- 4.5 = -9/40x + 9
- Rearranging gives: -9/40x = 4.5 - 9
- -9/40x = -4.5
- x = (-4.5 * -40) / 9 = 20
- Thus, P = (20, 4.5).
- To find point Q (intersection of EF and BD):
- Since EF is horizontal at y = 4.5, we already found Q at the same x-coordinate:
- Q = (20, 4.5).

6. **Length of Segment PQ**:
- The length of segment PQ can be calculated using the distance formula:
- PQ = |x2 - x1| = |20 - 20| = 0.
- However, since we need the length of PQ in terms of the rectangle's dimensions, we need to consider the division of BD by the midpoints:
- Since P and Q divide BD into three equal parts, we calculate the length of BD:
- Length of BD = √[(40 - 0)² + (0 - 9)²] = √(1600 + 81) = √1681 = 41 cm.
- Therefore, PQ = 41 cm / 3 = 13.67 cm.

### Final Answer:
The length of PQ is **13.67 cm**.