Midpoint Theorem

Q.

The line-segment joining the mid-points of any two sides of a triangle

1. Is parallel to the third side but it's dimension is not determinable

2. Is half as long as the third side but can't say if it is parallel to it

3. Is parallel to and half as long as the third side

4. None of the above

Q.

LMNO is a trapezium with $LM\parallel NO$. If P and Q are the mid - points of LO and MN respectively and LM =5 cm and ON =10 cm then PQ =

1. 2.5 cm

2. 5 cm

3. 7.5 cm

4. 15 cm

Q. In a triangle ABC, D and E are the midpoints of AB, and AC respectively. Also, DE is parallel to BC. If BC = 8 units, find the value of DE.

1. 1 unit
2. 2 units
3. 3 units
4. 4 units

Q. In the given figure, M, N and P are the mid-points of AB, AC and BC respectively. If MN = 3 cm, NP = 3.5 cm and MP = 2.5 cm, find the perimeter of the triangle ABC (in cm).

1. 18

Q. In the given figure, D and E are mid points of sides AB and AC respectively of $\Delta$ABC. Which of the following is true ?

1. $DE=\frac{BC}{2},\text{by mid point theorem}$
2. $DE=\frac{BC}{2},\text{by converse of mid point theorem}$
3. $DE=\frac{AB}{2},\text{by mid point theorem}$
4. $DE=\frac{AB}{2},\text{by converse of mid point theorem}$

Q. In this figure, triangle ABC is right - angled at B. Given that AB = 9 cm, AC = 15 cm and D and E are the mid-points of the sides AB and AC respectively, calculate the area of $\Delta ADE$.

1. $13c{m}^2$
2. $12c{m}^2$
3. $14c{m}^2$
4. $13.5c{m}^2$

Q.

If the circumradius of a right angled triangle is $10cm,$ then find the length of the hypotenuse.

1. $10cm$

2. $15cm$

3. $20cm$

4. $30cm$

Q. In the given figure, AC = 10 cm and AB = 8 cm. A perpendicular is drawn from the midpoint of the hypotenuse of the right triangle to the base. Find the length of DE.

1. 9 cm
2. 8 cm
3. 4 cm
4. 3 cm

Q.

In the given figure, ABCDE is a pentagon. A line through B parallel to AC meets DC produced at F. Which of the following is true.

1. area (AEDF) = area (ABCF)

2. area (DEAF) = area (ABCF)

3. area (ABCDE) = area(ABCF)

4. area (AEDF) = area (ABCDE)

Q. In the given figure, AD = DB and AE = EC. Then, $BC:DE=$___.

1. $1:2$
2. $2:1$
3. $1:4$
4. $4:1$

Q.

A quadrilateral ABCD is drawn in which the mid points of sides AB, BC, CD and AD are P, Q, R and S respectively.

If quadrilateral ABCD is a parallelogram, what can be said about sides SP and QR?

1. Equal and parallel

2. Equal

3. Parallel

4. Neither equal nor parallel

Q. In figure, seg PD is a median of PQR. Point T is the mid point of seg PD. Produced QT intersects PR at M. Show that $\frac{PM}{PR}=\frac{1}{3}.$ [Hint : Draw DN || QM.]

Q.

In the parallelogram, D is the mid point of side PQ. Which of the following options is NOT true?

1. PQ = RS

2. PD = $\frac{1}{2}$ RS

3. DQ = $\frac{1}{2}$RS

4. All of the above

Q.

In the given figure, D and E are mid points of sides AB and AC of $△$ABC. Which of the following is true ?

1. ​$DE=\frac{1}{2}BC$ , by converse of mid point theorem

2. $DE=\frac{1}{2}AB$, by converse of mid point theorem

3. $DE=\frac{1}{2}BC$, by mid point theorem

4. $DE=\frac{1}{2}AB$, by mid point theorem

Q. In the question, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice:

Assertion: In a parallelogram PQRS, area of △PQS is equal to area of △QSR.
Reason: A diagonal of a parallelogram divides it into two congruent triangles.

A. Both assertion (A) and reason (R) are true, but reason (R) is the correct explanation of assertion (A).
B. Both assertion (A) and reason (R) are true, but reason (R) is not the correct explanation of assertion (A).
C. Assertion (A) is true, but reason (R) is false.
D. Assertion (A) is false, but reason (R) is true.

Q.

In the given figure, if D and E are the mid points of sides AB and AC, respectively, and DE is produced to F such that E is the mid point of FD, which of the following options is correct?

1. ΔADE ≅ ΔCEF

2. ΔADE ≅ ΔCFE

3. ΔAED ≅ ΔCFE

4. ΔAED ≅ ΔFEC

Q. The heights of two buildings are 34m and 29m respectively. If the distance between the two building is 12m, find the distance between their tops.

1. $12m$
2. $13m$
3. $16m$
4. $14m$

Q.

In the given figure, E is the midpoint of side AC and D is the midpoint of side AB, then

S1 : DE = $\frac{1}{2}$BC and DE || BC

S2 : Midpoint theorem : A line segment joining midpoints of 2 sides of a triangle is parallel to the third side and is half of it.

1. S1 is true and S2 is false.

2. S1 is false and S2 is true.

3. S1 and S2 are both true.

4. S1 and S2 are both false.

Q.

In the given figure, $D$ and $E$ are mid points of sides $AB$ and $AC$ of $△ABC.$ Which of the following is correct?

1. $DE=\frac{1}{2}BC,$ by mid point theorem

2. $DE=\frac{1}{2}BC,$ by converse of mid point theorem

3. $DE=\frac{1}{2}AB,$ by mid point theorem

4. $DE=\frac{1}{2}AB,$ by converse of mid point theorem

Q.

In the figure given below, AC = 10 cm and AB = 8 cm. A perpendicular is drawn from the midpoint of the hypotenuse of the right triangle to the base. Find the length of DE.

1. 9 cm

2. 4 cm

3. 8 cm

4. 3 cm

Q.

In Fig. BM and DN are both perpendiculars to the segments AC and BM = DN. Prove that AC bisects BD [3 MARKS]

Q. P is the mid-point of the side CD of a parallelogram ABCD. A line through C parallel to PA intersects AB at Q and DA produced at R. Then:

1. DA = AR and CQ = QR
2. PA = AR
3. CR = DR
4. CR = 2CQ

Q. Construct the bisector of an angle ${75}^o$.

Q.

In triangle ABC, the medians BE and CF intersect at G. AD is a line meeting BC at D. If GD is 1.5 cm, then the value of AD is .

1. 3.5 cm
2. 4 cm
3. 4.5 cm

Q.

A quadrilateral, $ABCD$ is drawn in which the mid points of sides $AB,BC,CD$ and $AD$ are $P,Q,R$ and $S$ respectively.

If quadrilateral, $ABCD$ is a parallelogram, what can you say about the quadrilateral, $PQRS?$

1. Parallelogram

2. Rhombus

3. Rectangle

4. Can’t say

Q. If P and Q are the mid points of AB and CA respectively, then match the corresponding values of BC and PQ.

1. PQ = 8 cm
2. PQ = 7 cm
3. BC = 8 cm
4. BC = 18 cm

Q.

In the $\Delta ABC,\angle B$ is a right angle, $D$ and $E$ are the mid-points of the sides $AB=3cm$ and $AC=5cm$ respectively. Then, the length of $DE$ is

1. $4cm$

2. $2cm$

3. $5cm$

4. $3cm$

Q. Match the following with the value of $x$, if D and E are the mid points of side AB and AC respectively

1. 6
2. 14
3. 4

Q. Match the following with the value of $x$, if D and E are the mid points of side AB and AC respectively

1. 6
2. 14
3. 4

Q. For the following figure, prove that $AD=BC$ and $\angle ADB=\angle BCA$ (2 marks)