Question 6 of 14intermediate🔧 ApplyLong Answer3 marks

ABCD is a trapezium in which AB∥DCAB \parallel DC and its diagonals intersect each other at the point OO. Show that AOBO=CODO\dfrac{AO}{BO} = \dfrac{CO}{DO}.

Correct Answer

Given: ABCD is a trapezium with AB∥DCAB \parallel DC. Diagonals AC and BD intersect at OO.

To Prove:

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

Proof:

In ΔAOB\Delta AOB and ΔCOD\Delta COD:

∠OAB=∠OCD(Alternate angles, since AB∥DC and AC is transversal)\angle OAB = \angle OCD \quad \text{(Alternate angles, since } AB \parallel DC \text{ and AC is transversal)} ∠OBA=∠ODC(Alternate angles, since AB∥DC and BD is transversal)\angle OBA = \angle ODC \quad \text{(Alternate angles, since } AB \parallel DC \text{ and BD is transversal)} ∠AOB=∠COD(Vertically opposite angles)\angle AOB = \angle COD \quad \text{(Vertically opposite angles)}

Therefore, ΔAOB∼ΔCOD\Delta AOB \sim \Delta COD (AAA similarity criterion)

Since corresponding sides of similar triangles are in the same ratio:

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

Cross multiplying both sides:

AO×DO=CO×BOAO \times DO = CO \times BO

Therefore, AOBO=CODO\dfrac{AO}{BO} = \dfrac{CO}{DO}

Hence proved.

Exercise: EXERCISE 6.2 | Q: 9 | (Page Page 13)
For More Understanding

Explanation

This question tests the application of triangle similarity in trapeziums. The key insight is that when AB∥DCAB \parallel DC, the diagonals create two triangles (ΔAOB\Delta AOB and ΔCOD\Delta COD) that are similar by the AAA criterion.

The alternate angles are equal due to parallel lines, and vertically opposite angles are always equal. Once similarity is established, the proportionality of corresponding sides directly gives the required ratio.

The textbook context confirms this approach in Question 3, which asks to prove OAOC=OBOD\dfrac{OA}{OC} = \dfrac{OB}{OD} — the same relationship expressed differently.

Solution Steps

  1. Step 1: Identify triangles ΔAOB\Delta AOB and ΔCOD\Delta COD formed by intersecting diagonals

  2. Step 2: Prove similarity using AAA criterion (alternate angles from parallel lines, vertically opposite angles)

  3. Step 3: Apply the property that corresponding sides of similar triangles are proportional

  4. Step 4: Rearrange the proportion to get AOBO=CODO\dfrac{AO}{BO} = \dfrac{CO}{DO}

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