Question 7 of 14advanced🔧 ApplyLong Answer4 marks

The diagonals of a quadrilateral ABCD intersect each other at the point O such that AOBO=CODO\frac{AO}{BO} = \frac{CO}{DO}. Show that ABCD is a trapezium.

Correct Answer

Given: In quadrilateral ABCD, diagonals AC and BD intersect at O such that

AOBO=CODO\frac{AO}{BO} = \frac{CO}{DO}

To Prove: ABCD is a trapezium (i.e., AB || DC).

Construction: Join A to B, B to C, C to D, and D to A.

Proof:

We have

AOBO=CODO\frac{AO}{BO} = \frac{CO}{DO}

(Given)

Rearranging, we get

AOCO=BODO\frac{AO}{CO} = \frac{BO}{DO}

In ΔAOB and ΔCOD:

  • ∠AOB=∠COD\angle AOB = \angle COD (Vertically opposite angles)
  • AOCO=BODO\frac{AO}{CO} = \frac{BO}{DO} (From above)

Therefore, ΔAOB ~ ΔCOD (SAS similarity criterion)

From similarity of triangles:

∠OAB=∠OCD\angle OAB = \angle OCD

(Corresponding angles of similar triangles)

Now, ∠OAB\angle OAB and ∠OCD\angle OCD are alternate interior angles formed by transversal AC intersecting lines AB and DC.

Since alternate interior angles are equal, AB || DC.

Therefore, ABCD is a trapezium with AB || DC.

∴ Hence proved.

Exercise: EXERCISE 6.2 | Q: 10 | (Page Page 13)
For More Understanding

Explanation

This question is the converse of the property that in a trapezium with parallel sides, the diagonals divide each other proportionally. The key insight is to recognize that the given proportion and vertically opposite angles allow us to establish triangle similarity using SAS criterion. From the similarity, corresponding angles are equal, which gives us alternate interior angles equal — the condition for lines to be parallel. The textbook context shows similar problems (Questions 3 and 9) proving the forward direction, while Question 10 asks for the converse proof.

Solution Steps

  1. Step 1: Rearrange the given proportion to get AOCO=BODO\frac{AO}{CO} = \frac{BO}{DO}

  2. Step 2: Identify that ∠AOB=∠COD\angle AOB = \angle COD (vertically opposite angles)

  3. Step 3: Apply SAS similarity criterion to prove ΔAOB ~ ΔCOD

  4. Step 4: Use corresponding angles of similar triangles to get ∠OAB=∠OCD\angle OAB = \angle OCD

  5. Step 5: Recognize these as alternate interior angles and conclude AB || DC

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