Class 9 — Mathematics (NCERT)
Quadrilaterals: Properties of Parallelograms and Mid-point Theorem
Class 9
- ✓Define a quadrilateral and identify its basic components.
- ✓State and apply the properties of a parallelogram to solve problems.
- ✓State and apply the Mid-point Theorem and its converse to find lengths and prove parallelism.
- ✓Solve problems involving the properties of parallelograms and the Mid-point Theorem in various geometric figures.
Key concepts
A quadrilateral is a closed figure formed by four line segments. It has four sides, four angles, and four vertices. The sum of the interior angles of a quadrilateral is 360°.
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. If ABCD is a parallelogram, then AB || DC and AD || BC.
In a parallelogram, the following properties hold true:\n1. Opposite sides are equal in length (AB = DC and AD = BC).\n2. Opposite angles are equal in measure (∠A = ∠C and ∠B = ∠D).\n3. Diagonals bisect each other (if diagonals AC and BD intersect at O, then AO = OC and BO = OD).\n4. Consecutive angles are supplementary (∠A + ∠B = 180°, ∠B + ∠C = 180°, etc.).
The line segment joining the mid-points of two sides of a triangle is parallel to the third side and is half of it. If D and E are the mid-points of sides AB and AC respectively of ΔABC, then DE || BC and DE = (1/2)BC.
A line drawn through the mid-point of one side of a triangle parallel to another side bisects the third side. If D is the mid-point of side AB of ΔABC and a line is drawn through D parallel to BC, intersecting AC at E, then E is the mid-point of AC.
Key facts to remember
- 1A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
- 2In a parallelogram, opposite sides are equal.
- 3In a parallelogram, opposite angles are equal.
- 4In a parallelogram, diagonals bisect each other.
- 5In a parallelogram, consecutive angles are supplementary.
- 6The Mid-point Theorem states that the line segment joining the mid-points of two sides of a triangle is parallel to the third side and is half of it.
- 7The converse of the Mid-point Theorem states that a line through the mid-point of one side of a triangle parallel to another side bisects the third side.
Worked examples
Example 1
In parallelogram ABCD, ∠A = 75°. Find the measures of ∠B, ∠C, and ∠D.
Answer
∠B = 105°, ∠C = 75°, ∠D = 105°.
Always state the property used for each step in your solution.
Example 2
In ΔPQR, S and T are the mid-points of sides PQ and PR respectively. If QR = 12 cm, find the length of ST.
Answer
The length of ST is 6 cm.
The Mid-point Theorem also implies ST || QR, which can be useful in other problems.
Example 3
ABCD is a quadrilateral in which P, Q, R, and S are mid-points of sides AB, BC, CD, and DA respectively. Show that PQRS is a parallelogram.
Answer
PQRS is a parallelogram.
This is a standard proof demonstrating the application of the Mid-point Theorem.
Common mistakes
- ✗Confusing properties of a parallelogram with those of other quadrilaterals (e.g., assuming diagonals are equal).
- ✗Incorrectly applying the Mid-point Theorem by not ensuring both points are mid-points or by misidentifying the third side.
- ✗Assuming a figure is a parallelogram without proving it first, leading to incorrect deductions.
- ✗Making calculation errors when finding angles or lengths based on properties.
Exam tips
- ★Always draw a neat and labelled diagram for each problem to visualise the given information.
- ★Clearly state the property or theorem you are using at each step of your solution.
- ★For proofs, write down the 'Given', 'To Prove', and 'Construction' (if any) clearly before starting the proof.
- ★Practice a variety of problems, including those that combine different properties and theorems, to build confidence.
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