Question:
In the given figure, AB $\parallel$ DC. If OB = 3OD and CD = 1.8 cm, then find the length AB.
In the given figure, AB $\parallel$ DC. If OB = 3OD and CD = 1.8 cm, then find the length AB.
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Whenever you have a trapezium with parallel bases, the triangles formed by the segments of the diagonals are always similar.
This means the ratio of the bases is equal to the ratio of the diagonal segments: \(AB / CD = OB / OD\).
This means the ratio of the bases is equal to the ratio of the diagonal segments: \(AB / CD = OB / OD\).
Updated On: Jun 25, 2026
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Correct Answer: 5.4
Solution and Explanation
Step 1: Understanding the Question:
This question is from Triangles, specifically dealing with similar triangles formed by parallel lines intersecting in a trapezium-like shape.
We are given that \(AB \parallel DC\).
The diagonals \(AC\) and \(BD\) intersect at point \(O\). We are given the relation \(OB = 3OD\) and the side length \(CD = 1.8\text{ cm}\). We need to determine the length of the parallel side \(AB\).
Step 2: Key Formula or Approach:
Since \(AB \parallel DC\), the triangles formed by the diagonals, \(\triangle AOB\) and \(\triangle COD\), are similar by AAA similarity criterion:
1. \(\angle AOB = \angle COD\) (vertically opposite angles)
2. \(\angle OAB = \angle OCD\) (alternate interior angles)
3. \(\angle OBA = \angle ODC\) (alternate interior angles)
Thus, \(\triangle AOB \sim \triangle COD\).
For similar triangles, the ratio of corresponding sides is equal: \[ \frac{AB}{CD} = \frac{OB}{OD} \]
Step 3: Detailed Explanation:
1. Establish similarity between \(\triangle AOB\) and \(\triangle COD\):
In \(\triangle AOB\) and \(\triangle COD\): - \(\angle AOB = \angle COD\) (vertically opposite angles)
- \(\angle OAB = \angle OCD\) (since \(AB \parallel DC\) and \(AC\) is a transversal, alternate interior angles are equal)
Therefore, by AA similarity criterion: \[ \triangle AOB \sim \triangle COD \] 2. Write down the ratio of their corresponding sides: \[ \frac{AB}{CD} = \frac{OB}{OD} \] 3. We are given the relation: \[ OB = 3OD \implies \frac{OB}{OD} = 3 \] 4. Substitute the ratio \(\frac{OB}{OD} = 3\) and \(CD = 1.8\text{ cm}\) into the side-ratio equation: \[ \frac{AB}{1.8} = 3 \] 5. Solve for \(AB\) by multiplying both sides by 1.8: \[ AB = 3 \times 1.8 \] \[ AB = 5.4\text{ cm} \]
Step 4: Final Answer:
The length of \(AB\) is 5.4 cm.
This question is from Triangles, specifically dealing with similar triangles formed by parallel lines intersecting in a trapezium-like shape.
We are given that \(AB \parallel DC\).
The diagonals \(AC\) and \(BD\) intersect at point \(O\). We are given the relation \(OB = 3OD\) and the side length \(CD = 1.8\text{ cm}\). We need to determine the length of the parallel side \(AB\).
Step 2: Key Formula or Approach:
Since \(AB \parallel DC\), the triangles formed by the diagonals, \(\triangle AOB\) and \(\triangle COD\), are similar by AAA similarity criterion:
1. \(\angle AOB = \angle COD\) (vertically opposite angles)
2. \(\angle OAB = \angle OCD\) (alternate interior angles)
3. \(\angle OBA = \angle ODC\) (alternate interior angles)
Thus, \(\triangle AOB \sim \triangle COD\).
For similar triangles, the ratio of corresponding sides is equal: \[ \frac{AB}{CD} = \frac{OB}{OD} \]
Step 3: Detailed Explanation:
1. Establish similarity between \(\triangle AOB\) and \(\triangle COD\):
In \(\triangle AOB\) and \(\triangle COD\): - \(\angle AOB = \angle COD\) (vertically opposite angles)
- \(\angle OAB = \angle OCD\) (since \(AB \parallel DC\) and \(AC\) is a transversal, alternate interior angles are equal)
Therefore, by AA similarity criterion: \[ \triangle AOB \sim \triangle COD \] 2. Write down the ratio of their corresponding sides: \[ \frac{AB}{CD} = \frac{OB}{OD} \] 3. We are given the relation: \[ OB = 3OD \implies \frac{OB}{OD} = 3 \] 4. Substitute the ratio \(\frac{OB}{OD} = 3\) and \(CD = 1.8\text{ cm}\) into the side-ratio equation: \[ \frac{AB}{1.8} = 3 \] 5. Solve for \(AB\) by multiplying both sides by 1.8: \[ AB = 3 \times 1.8 \] \[ AB = 5.4\text{ cm} \]
Step 4: Final Answer:
The length of \(AB\) is 5.4 cm.
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