ABCD is a quadrilateral in which AB || DC, and E and F are the mid points of the diagonals AC and BD, respectively. If AB = 12 cm, BC = 40 cm, DC = 30 cm, and AD = 45 cm, then what is the length (in cm) of EF?
Understanding the Quadrilateral Geometry Problem
We are given a quadrilateral ABCD where one pair of opposite sides, AB and DC, are parallel (AB || DC). This specific type of quadrilateral is known as a trapezoid.
We are also given that E is the midpoint of the diagonal AC, and F is the midpoint of the diagonal BD. Our goal is to find the length of the line segment EF connecting these two midpoints.
The given lengths are:
- AB = 12 cm
- BC = 40 cm
- DC = 30 cm
- AD = 45 cm
Note: Since AB || DC, the lengths of the non-parallel sides BC and AD are not required to find the length of EF.
Geometric Property for Trapezoid Midpoints
There is a specific theorem related to the segment connecting the midpoints of the diagonals of a trapezoid. If E and F are the midpoints of the diagonals AC and BD, respectively, of a trapezoid ABCD with AB || DC, then the length of the segment EF is half the absolute difference between the lengths of the parallel sides.
The formula is:
$ EF = \frac{1}{2} |DC - AB| $
Where |...| denotes the absolute value.
Calculating the Length of EF
Now, we can use the given lengths of the parallel sides, AB and DC, in the formula:
- Identify parallel sides: AB = 12 cm and DC = 30 cm.
- Apply the formula: Substitute the values into the formula for EF. $ EF = \frac{1}{2} |30 \text{ cm} - 12 \text{ cm}| $
- Calculate the difference: Find the difference between the lengths of DC and AB. $ EF = \frac{1}{2} |18 \text{ cm}| $
- Calculate EF: Take half of the absolute difference. $ EF = \frac{1}{2} \times 18 \text{ cm} $ $ EF = 9 \text{ cm} $
Conclusion
Therefore, the length of the segment EF connecting the midpoints of the diagonals AC and BD is 9 cm.
Important Questions from Geometry
- ABCD is a trapezium in which BC || AD and AC = CD. If $\angle ABC = 24^\circ$ and $\angle BAC = 112^\circ$, then what is the measure of $\angle ACD$ (in degree)?
- ABCD is a trapezium in which BC || AD and AC = CD. If $\angle ABC = 20^{\circ}$ and $\angle BAC = 131^{\circ}$, then what is the measure of $\angle ACD$ (in degree)?
- A regular polygon has 30 sides. Find the measure of one interior angle.
- ABCD is a trapezium in which BC $||$ AD and AC = CD. If $\angle ABC = 19^{\circ}$ and $\angle BAC = 137^{\circ}$, then what is the measure of $\angle ACD$ (in degree)?
- Find the number of sides of a regular polygon, each of whose interior angles measures $162^\circ$.
