Question 8 of 14intermediate🔧 ApplyLong Answer3 marks

Diagonals ACAC and BDBD of a trapezium ABCDABCD with AB∥DCAB \parallel DC intersect each other at the point OO. Using a similarity criterion for two triangles, show that OAOC=OBOD\dfrac{OA}{OC} = \dfrac{OB}{OD}.

Correct Answer

In trapezium ABCDABCD, AB∥DCAB \parallel DC and diagonals ACAC and BDBD intersect at OO. We consider triangles ΔOAB\Delta OAB and ΔOCD\Delta OCD.

Since AB∥DCAB \parallel DC and ACAC acts as a transversal, the alternate interior angles are equal, so ∠OAB=∠OCD\angle OAB = \angle OCD. Similarly, with BDBD as a transversal, ∠OBA=∠ODC\angle OBA = \angle ODC. Additionally, ∠AOB=∠COD\angle AOB = \angle COD because they are vertically opposite angles.

By the AAA similarity criterion (Angle-Angle-Angle), since all corresponding angles are equal, ΔOAB∼ΔOCD\Delta OAB \sim \Delta OCD. When two triangles are similar, their corresponding sides are in the same ratio. Therefore,

OAOC=OBOD=ABDC\frac{OA}{OC} = \frac{OB}{OD} = \frac{AB}{DC}

Hence, it is shown that OAOC=OBOD\dfrac{OA}{OC} = \dfrac{OB}{OD}.

Exercise: EXERCISE 6.3 | Q: 3 | (Page Page 23)
For More Understanding

Explanation

This question corresponds to Question 9 in the provided context. The solution relies on establishing the similarity between ΔOAB\Delta OAB and ΔOCD\Delta OCD.

Because ABAB is parallel to DCDC, the angles formed by the intersecting diagonals (acting as transversals) are equal (alternate interior angles) or vertically opposite. The context explicitly defines the criteria for similarity: corresponding angles must be equal.

Once similarity is proven (AAA), the property that corresponding sides are in proportion is used to derive the required ratio.

Solution Steps

  1. Step 1: Identify ΔOAB\Delta OAB and ΔOCD\Delta OCD within trapezium ABCDABCD where AB∥DCAB \parallel DC.

  2. Step 2: Show that ∠OAB=∠OCD\angle OAB = \angle OCD and ∠OBA=∠ODC\angle OBA = \angle ODC (Alternate Interior Angles) and ∠AOB=∠COD\angle AOB = \angle COD (Vertically Opposite Angles).

  3. Step 3: Conclude ΔOAB∼ΔOCD\Delta OAB \sim \Delta OCD by AAA similarity criterion.

  4. Step 4: Apply the property of similar triangles that corresponding sides are proportional:

    OAOC=OBOD\frac{OA}{OC} = \frac{OB}{OD}
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