468 + Interview Questions in QUADRILATERALS IN GENERAL AWARENESS Page 10 InterviewSolution

451.	A quadrilateral in which one pair of opposite sides are parallel is a A) parallelogram B) kite C) trapeziumD) rhombus
Answer» Correct option is (C) trapezium A quadrilateral in which one pair of opposite sides are parallel is a trapezium. C) trapezium

451.

A quadrilateral in which one pair of opposite sides are parallel is a A) parallelogram B) kite C) trapeziumD) rhombus

Answer»

Correct option is (C) trapezium

A quadrilateral in which one pair of opposite sides are parallel is a trapezium.

C) trapezium

452.	If diagonal of a parallelogram bisects one of the angles of theparallelogram, it also bisects the second angle. Also, prove that it is arhombus.
Answer» Given- AC diagonal bisects `/_A` AB\|\|CD and AC is transverse `/_3=/_1-(1)` AD\|\|BC and AC is transverse `/_2=/_4-(2)` AC bisects`/_A` `/_1=/_2-(3)` `/_3=/_4` AC bisects `/_A` `/_1=/_4` AB=DC and BC=AB AB=BC-CD=CD ABCD is a rhombus.

Discussion

453.	In Figure, `M ,N`and `P`are the mid-points of `A B ,A C`and `B C`respectively. If `M N=3c m ,N P=3. 5c m`and `M P=2. 5c m ,`calculate `B C ,A B`and `A Cdot`
Answer» In`/_ABC` Since, H is the mid point of AB and N is the mid point of AC MN\|\|BC and `MN=1/2BC` BC=2MN P is midpoint of BC PN\|\|AB and `PN=1/2AB` NP\|\|AC and `MP=1/2AC` BC=23=6cm AB=23.5=7cm AC=2*2.5=5cm.

Discussion

454.	In ` A B C ,A D`is the median through `A`and `E`is the mid-point of `A D`. `B E`produced meets `A C`in `F`(Figure). Prove that `A F=1/3A Cdot`
Answer» Construction DK\|\|BF In`/_ADK` E is the midpoint od AD and EF\|\|DK F is the midpoint of AK AF=FK-(1) K is the mid point of FC FK=KC-(2) FK=KC=KC AC=AF+FK+KC AC=AF+AF+AF=3AF `AF=1/3AC`.

Discussion

455.	In a ` A B C`, find the measures of the angles of the triangle formed by joining themid-points of the sides of this triangle.
Answer» Given D,E,F are mid point AB,BC and CA DE\|\|AB,FE\|\|BC and DF\|\|CA DE\|\|AB and BC and CA tranverse `/_CDE=/_B` `/_CED=/_A` EF\|\|BC `/_AEF=/_C` and `/_AFE=/_B` OF\|\|CA `/_BDF=/_C` and `/_BFD=/_A` `/_BDF+/_FDE+/_EDC=180^@` `/_C+/_FOE+/_B=180^@` `/_FDE=180-(/_B+/_C)` `/_FDE=/_A` `/_DEF=180^@-(/_A+/_C)=/_B` `/_EFD=180^@-(/_A+/_B)=/_A` `/_D=/_A,/_E=/_B,/_F=/_C`.

Discussion

456.	Fill in the blanks to make the following statements correct :(i) The triangle formed by joining the mid-points of the sides of an isosceles triangle is….. (ii) The triangle formed by joining the mid-points of the sides of a right triangle is….. (iii) The figure formed by joining the mid-points of consecutive sides of a quadrilateral is……
Answer» (i) Isosceles. (ii) Right triangle. (iii) Parallelogram.

456.

Fill in the blanks to make the following statements correct :(i) The triangle formed by joining the mid-points of the sides of an isosceles triangle is….. (ii) The triangle formed by joining the mid-points of the sides of a right triangle is….. (iii) The figure formed by joining the mid-points of consecutive sides of a quadrilateral is……

Answer»

(i) Isosceles.

(ii) Right triangle.

(iii) Parallelogram.

Discussion

457.	Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.
Answer» Data : P, Q, R and S are mid-points of AB, BC, CD and DA respectively in quadrilateral ABCD. To Prove: PR and SQ line segments bisect mutually. Construction: Join the diagonal AC. Proof: In ∆ADC, S and R are the mid-points of AD and DC. ∴ SR \|\| AC SR = \(\frac{1}{2}\)AC ………… (i) (Mid-point Theorem) Similarly, in ∆ABC, PQ \|\| AC PQ = \(\frac{1}{2}\)AC …………. (ii) From (i) and (ii), SR = PQ and SR \|\| PQ ∴ PQRS is a parallelogram. PR and SQ are the diagonals of parallelogram PQRS. ∴PR and SQ meet at O’.

457.

Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

Answer»

Data : P, Q, R and S are mid-points of AB, BC, CD and DA respectively in quadrilateral ABCD.

To Prove: PR and SQ line segments bisect mutually.

Construction: Join the diagonal AC.

Proof: In ∆ADC, S and R are the mid-points of AD and DC.

∴ SR || AC

SR = \(\frac{1}{2}\)AC ………… (i)

(Mid-point Theorem)

Similarly, in ∆ABC,

PQ || AC

PQ = \(\frac{1}{2}\)AC …………. (ii)

From (i) and (ii),

SR = PQ and

SR || PQ

∴ PQRS is a parallelogram. PR and SQ are the diagonals of parallelogram PQRS.

∴PR and SQ meet at O’.

Discussion

458.	Show that the line segments joining the mid points of the opposite sides of a quadrilateral and bisect each other.
Answer» Let ABCD be a quadrilateral. P, Q, R, S are the midpoints of sides of □ABCD. Join (P, Q), (Q, R), (R, S) and (S, P). In ΔABC; P, Q are the midpoints of AB and BC. ∴ PQ // AC and PQ = 1/2 AC ………….(1) Also from ΔADC S, R are the midpoints of AD and CD SR // AC and SR = 1/2 AC …………………(2) ∴ From (1) & (2) PQ = SR and PQ //SR ∴ □PQRS is a parallelogram. Now PR and QS are the diagonals of □ PQRS. ∴ PR and QS bisect each other.

458.

Show that the line segments joining the mid points of the opposite sides of a quadrilateral and bisect each other.

Answer»

Let ABCD be a quadrilateral.

P, Q, R, S are the midpoints of sides of □ABCD.

Join (P, Q), (Q, R), (R, S) and (S, P).

In ΔABC; P, Q are the midpoints of AB and BC.

∴ PQ // AC and PQ = 1/2 AC ………….(1)

Also from ΔADC

S, R are the midpoints of AD and CD

SR // AC and SR = 1/2 AC …………………(2)

∴ From (1) & (2)

PQ = SR and PQ //SR

∴ □PQRS is a parallelogram.

Now PR and QS are the diagonals of □ PQRS.

∴ PR and QS bisect each other.

Discussion

459.	The quadrilateral in which the diagonals are equal but not perpendicular to each other is A) Square B) Rhombus C) Parallelogram D) Rectangle
Answer» Correct option is (D) Rectangle The quadrilateral in which the diagonals are equal but not perpendicular to each other is a rectangle. Correct option is D) Rectangle

459.

The quadrilateral in which the diagonals are equal but not perpendicular to each other is A) Square B) Rhombus C) Parallelogram D) Rectangle

Answer»

Correct option is (D) Rectangle

The quadrilateral in which the diagonals are equal but not perpendicular to each other is a rectangle.

Correct option is D) Rectangle

Discussion

460.	The quadrilateral in which the diagonals bisect each other and they are perpendicular to each other A) Rectangle B) Rhombus C) Parallelogram D) Trapezium
Answer» Correct option is: B) Rhombus

Discussion

461.	What is Polygons?
Answer» We are familiar with plane figures bounded by straight line segments as sides. They are known as Polygons.

Discussion

462.	The three angles of a quadrilateral measure 56°, 100° and 88°. Find the measure of the fourth angle.
Answer» Let the measure of the fourth angle be x. As we know that sum of the angles of a quadrilateral is 360°. 56° + 100° + 88° + x = 360° ⇒ 244° + x = 360° ⇒ 360° – 244° = 116° Hence, the measure of the fourth angle is 116°.

462.

The three angles of a quadrilateral measure 56°, 100° and 88°. Find the measure of the fourth angle.

Answer»

Let the measure of the fourth angle be x.

As we know that sum of the angles of a quadrilateral is 360°.

56° + 100° + 88° + x = 360°

⇒ 244° + x = 360°

⇒ 360° – 244° = 116°

Hence, the measure of the fourth angle is 116°.

Discussion

463.	If the degree measures of the angles of quadrilateral are 4x, 7x, 9x and 10x, what is the sum of the measures of the smallest angle and largest angle?A. 140° B. 150° C. 168° D. 180°
Answer» Option : (C) Given, ABCD is a parallelogram, Angles of quadrilateral 4x , 7x , 9x , 10x = 4x+7x+9x+10x = 360° [angle sum property of quadrilateral] = 30x = 360° = x = \(\frac{360°}{30}\) = 12° Hence, Sum of smallest and largest angles = 4x+10x = 4×12+10×12 = 48°+120° = 168°

463.

If the degree measures of the angles of quadrilateral are 4x, 7x, 9x and 10x, what is the sum of the measures of the smallest angle and largest angle?A. 140° B. 150° C. 168° D. 180°

Answer»

Option : (C)

Given,

ABCD is a parallelogram,

Angles of quadrilateral 4x , 7x , 9x , 10x

= 4x+7x+9x+10x = 360°

[angle sum property of quadrilateral]

= 30x = 360°

= x = \(\frac{360°}{30}\) = 12°

Hence,

Sum of smallest and largest angles = 4x+10x

= 4×12+10×12

= 48°+120° = 168°

Discussion

464.	How many diagonals does each of the following have?(a) A convex quadrilateral(b) A regular hexagon(c) A triangle
Answer» (a) Two, (b) 9, (c) 0 (zero)

Discussion

465.	What is the sum of the measures of the angles of a convex quadrilateral? Why this property hold if the quadrilateral is not convex? (Make a non-convex quadrilateral and try!)
Answer» Solution: Angle sum of a convex quadrilateral = (4 – 2) x 180⁰ = 2 x 180⁰ = 360⁰ Since, quadrilateral, which is not convex, i.e. concave has same number of sides i.e. 4 as a convex quadrilateral have, thus, a quadrilateral which not convex also hold this property. i.e. angle some of a concave quadrilateral is also equal to 360⁰

Discussion

466.	In the given Q figure, ABCD is a parallelogram in which ∠ADC = 130°, then ∠CBE is equal to:(A) 60°(B) 130°(C) 50°(D) 70°
Answer» Answer is (C) 50°

Discussion

467.	In figure, ABCD is a parallelogram then the value of x is:(A) 25°(B) 60°(C) 75°(D) 45°
Answer» Answer is (D) 45°

Discussion

468.	In the given figure, PQR is a triangle in which X and Y are the mid-point of the sides PR and QR respectively. If PQ = 6 cm, QR = 7 cm and PR = 8 cm, then XY is equal to:(A) 12 cm(B) 3.5 cm(C) 3 cm(D) 4 cm
Answer» Answer is (C) 3 cm

Discussion

Explore topic-wise InterviewSolutions in .

A quadrilateral in which one pair of opposite sides are parallel is a A) parallelogram B) kite C) trapeziumD) rhombus

If diagonal of a parallelogram bisects one of the angles of theparallelogram, it also bisects the second angle. Also, prove that it is arhombus.

In Figure, `M ,N`and `P`are the mid-points of `A B ,A C`and `B C`respectively. If `M N=3c m ,N P=3. 5c m`and `M P=2. 5c m ,`calculate `B C ,A B`and `A Cdot`

In ` A B C ,A D`is the median through `A`and `E`is the mid-point of `A D`. `B E`produced meets `A C`in `F`(Figure). Prove that `A F=1/3A Cdot`

In a ` A B C`, find the measures of the angles of the triangle formed by joining themid-points of the sides of this triangle.

Show that the line segments joining the mid-points of the opposite sides of a quadrilateral bisect each other.

Show that the line segments joining the mid points of the opposite sides of a quadrilateral and bisect each other.

The quadrilateral in which the diagonals are equal but not perpendicular to each other is A) Square B) Rhombus C) Parallelogram D) Rectangle

The quadrilateral in which the diagonals bisect each other and they are perpendicular to each other A) Rectangle B) Rhombus C) Parallelogram D) Trapezium

What is Polygons?

The three angles of a quadrilateral measure 56°, 100° and 88°. Find the measure of the fourth angle.

If the degree measures of the angles of quadrilateral are 4x, 7x, 9x and 10x, what is the sum of the measures of the smallest angle and largest angle?A. 140° B. 150° C. 168° D. 180°

How many diagonals does each of the following have?(a) A convex quadrilateral(b) A regular hexagon(c) A triangle

What is the sum of the measures of the angles of a convex quadrilateral? Why this property hold if the quadrilateral is not convex? (Make a non-convex quadrilateral and try!)

In the given Q figure, ABCD is a parallelogram in which ∠ADC = 130°, then ∠CBE is equal to:(A) 60°(B) 130°(C) 50°(D) 70°

In figure, ABCD is a parallelogram then the value of x is:(A) 25°(B) 60°(C) 75°(D) 45°

In the given figure, PQR is a triangle in which X and Y are the mid-point of the sides PR and QR respectively. If PQ = 6 cm, QR = 7 cm and PR = 8 cm, then XY is equal to:(A) 12 cm(B) 3.5 cm(C) 3 cm(D) 4 cm