Assertion (A): For a Δ ABC, line segment EF is drawn such that E is the midpoint of AB and F is a midpoint of AC. Then the quadrilateral formed EFCB is a trapezium.
Reason (R): The line segment joining the midpoint of two sides of a triangle is parallel to the third side.
Assertion (A) is true, Reason (R) is false.
Assertion (A) is false, Reason (R) is true.
Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).
Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).
Answer
According to the Midpoint Theorem:
"The line segment joining the midpoints of any two sides of a triangle is parallel to the third side and is equal to half of its length."
∴ Reason (R) is true.
In △ABC:
⇒ E is the midpoint of AB.
⇒ F is the midpoint of AC.
The line segment joining E and F is EF.
Therefore, by the Midpoint Theorem, EF ∥ BC.
The quadrilateral EFCB has one pair of parallel sides (EF and BC).
Therefore, the quadrilateral EFCB is indeed a trapezium.
∴ Assertion (A) is true.
∴ Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason (or explanation) for Assertion (A).
Hence, option 3 is the correct option.
Assertion (A): In a Δ DEF, we have DE = EF = DF = 6 cm. A line segment PQ is drawn parallel to DF such that EP = 3 cm. Then we can conclude that PQ = 3 cm.
Reason (R): Any line segment drawn inside a triangle parallel to the base of the triangle cuts the removing two sides in half.
Assertion (A) is true, Reason (R) is false.
Assertion (A) is false, Reason (R) is true.
Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).
Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).
Answer
According to converse of mid-point theorem, the straight line drawn through the mid-point of one side of a triangle parallel to another bisects the third side.
It is given that PQ is drawn parallel to DF. And, EP = 3 cm.
∴ P is midpoint of ED.
Therefore, Q will also be mid-point of EF.
According to Midpoint Theorem states that the line segment joining the midpoints of any two sides of a triangle is parallel to the third side and is equal to half of it.
PQ = x DF = x 6 cm
Thus, PQ = 3 cm.
∴ Assertion (A) is true.
A line segment parallel to the base does not necessarily cut the other two sides in half unless it passes through the midpoints.
∴ Reason (R) is false.
∴ Assertion (A) is true, Reason (R) is false.
Hence, option 1 is the correct option.
Assertion (A): Refer to the adjoining figure. Three lines p, q, r are parallel to each other and PQ = QR = 1 cm. Then we conclude that AB = AC.
Reason (R): If a tranversal makes equal intercepts on three parallel lines, then another transversal will also make equal intercepts.
Assertion (A) is true, Reason (R) is false.
Assertion (A) is false, Reason (R) is true.
Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason for Assertion (A).
Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct reason (or explanation) for Assertion (A).
Answer
According to equal intercept theorem, if a tranversal makes equal intercepts on three or more parallel lines, then any other line cutting them will also make equal intercepts.
∴ Reason (R) is true.
Three lines p, q, r are parallel to each other
PQ = QR = 1 cm.
⇒ PQ = QR
By equal intercept theorem.
⇒ AB = BC
From figure, AC = AB + BC
⇒ AC = AB + AB
⇒ AC = 2AB
⇒ AB = AC.
∴ Assertion (A) is true.
∴ Both Assertion (A) and Reason (R) are true, and Reason (R) is the correct reason (or explanation) for Assertion (A).
Hence, option 3 is the correct option.