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In the diagram, ABCD is a parallelogram with AB = 14 units - NSC Mathematics - Question 9 - 2023 - Paper 2
Question 9
In the diagram, ABCD is a parallelogram with AB = 14 units. AD is produced to E such that AD : DE = 4 : 3. EB intersects DC in F. EB = 21 units. 9.1 Calculate, with... show full transcript
Worked Solution & Example Answer:In the diagram, ABCD is a parallelogram with AB = 14 units - NSC Mathematics - Question 9 - 2023 - Paper 2
Step 1
Calculate, with reasons, the length of FB.
Answer
To find the length of FB, we use the property of ratios in parallel lines. We know that:
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The segments AD and DE are in the ratio given by AD : DE = 4 : 3. Hence, the total length of AD + DE is proportional.
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Since EB = 21 units and AD + DE = 4 + 3 = 7 parts, each part equals:
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Therefore, the lengths are:
- FB = 4 parts = 4 × 3 = 12 units.
Step 2
Prove, with reasons, that \( \Delta EDF \parallel \Delta AEB. \)
Answer
To prove that ( \Delta EDF \parallel \Delta AEB ):
- We know that ( E ) is common to both triangles.
- By the Corresponding Angles Postulate:
- Angle EDF corresponds to angle EAB (as they are alternate interior angles when a transversal intersects the two parallels).
- Also, angle EFD corresponds to angle EBA.
- Since two pairs of angles are equal, we conclude that: ( \Delta EDF \parallel \Delta AEB )
Step 3
Calculate, with reasons, the length of FC.
Answer
To determine the length of FC:
- We use the property of the corresponding sides of similar triangles:
- With DE || AB and DF = 6 units (from previous calculations), we set up the proportion:
- From the parallelogram properties, we know that the length of BC is equal to AD = 14 units.
- Thus:
- FC can be calculated from the triangle similarity (where BC = AD):
Final calculation gives us FC = 8 units.