Mid-Point Theorem & Intercept Theorem — Assertion-Reason Type Questions

Assertion (A): In the figure, if AD = DC = 4 cm, EC = 10 cm and DE || AB, then CE = 5 cm.

Reason (R): The straight line drawn through the mid-point of one side of a triangle parallel to other, bisects the third side.

1. A is true, R is false

2. A is false, R is true

3. Both A and R are true

4. Both A and R are false

Answer

By converse of mid-point theorem,

A line drawn through the midpoint of one side of a triangle, and parallel to another side, will bisect the third side.

∴ Reason (R) is true.

In △ABC,

Since, D is the mid-point of AC and DE // AB, thus C is mid-point of BC.

CE = BE

From figure,

BC = CE + BE = CE + CE = 2 CE

⇒ CE = $\dfrac{1}{2}BC = \dfrac{1}{2}$ × 10 = 5 cm.

∴ Assertion (A) is true.

Hence, option 3 is the correct option.

Assertion (A): The mid-points of the sides of a quadrilateral ABCD are joined in order to get quadrilateral PQRS. PQRS is a rhombus.

Reason (R): Adjacent sides of a rhombus are equal and perpendicular to each other.

1. A is true, R is false

2. A is false, R is true

3. Both A and R are true

4. Both A and R are false

Answer

Join AC and BD.

By mid-point theorem,

The line segment joining the mid points of any two sides of a triangle is parallel to the third side and is equal to half of it.

In △ABC,

P and Q are midpoints of AB and BC respectively.

∴ PQ || AC and PQ = $\dfrac{1}{2}$ AC (By midpoint theorem) .....(1)

Similarly in △ADC,

S and R are midpoints of AD and CD respectively.

∴ RS || AC and RS = $\dfrac{1}{2}$ AC (By midpoint theorem) .....(2)

In △ABD,

P and S are midpoints of AB and AD respectively.

∴ PS || BD and PS = $\dfrac{1}{2}$ BD (By midpoint theorem) .....(3)

Similarly in △BCD,

Q and R are midpoints of BC and CD respectively.

∴ QR || BD and QR = $\dfrac{1}{2}$ BD (By midpoint theorem) .....(4)

From (1) and (2) we get,

PQ = RS and PQ || RS

From (3) and (4) we get,

PS = QR and PS || QR

Since, opposite sides are parallel and equal.

Thus, PQRS is a parallelogram.

∴ Assertion (A) is false.

In rhombus, adjacent sides are equal and adjacent angles are supplementary.

∴ Reason (R) is false.

Hence, option 4 is the correct option.