- A5 cm
- B10 cm
- C2.5 cm
- D7.5 cm
Step-by-Step Solution
Converse of mid-point theorem: line through midpoint parallel to third side bisects the third. AE = .
Answer: A) 5 cm
In ΔABC, D is midpoint of AB. A line through D parallel to BC meets AC at E. If AC = 10 cm, find AE.
Class 9QuadrilateralsGeometry — Quadrilaterals
Key idea: The mid-point theorem
**Theorem (Mid-Point)**: The line segment joining the mid-points of any two sides of a triangle is parallel to the third side and equal to half of it.
In ΔABC, let D be midpoint of AB and E be midpoint of AC. Then DE ∥ BC and DE = (1/2) × BC.
**Converse**: A line through the midpoint of one side parallel to another side will bisect the third.
In ΔABC, if D is midpoint of AB and a line through D parallel to BC meets AC at E, then E is the midpoint of AC.
**Use**: this theorem solves a huge class of problems where you have midpoints of triangle sides — you can immediately conclude parallelism and half-length.
In ΔABC, let D be midpoint of AB and E be midpoint of AC. Then DE ∥ BC and DE = (1/2) × BC.
**Converse**: A line through the midpoint of one side parallel to another side will bisect the third.
In ΔABC, if D is midpoint of AB and a line through D parallel to BC meets AC at E, then E is the midpoint of AC.
**Use**: this theorem solves a huge class of problems where you have midpoints of triangle sides — you can immediately conclude parallelism and half-length.
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