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# NMTC Geometry Problems and Solutions

### NMTC 2019 Stage 1 Inter Question 5

The area of the curve enclosed by is :

(A) 16
(B) 12
(C) 8
(D) 4

Area of is equivalent to area of

So it is a rhombus

Hence area will be,

### NMTC 2019 Inter Stage 1 Question 11

In a rectangle , point lies on such that and point lies on such that 2. Lines and intersect at and respectively. If , are relatively prime positive integers, then the minimum value of is :

(i)
.(ii)

### NMTC 2019 Inter Stage 1 Question 13

A regular polygon has 100 sides each of length. A another regular polygon has 200 sided each of length 2. When the area of the larger polygon is divided by the area of the smaller polygon, the quotient is closest to the integer
(A) 2
(B) 4
(C) 8
(D) 16

Area of polygon 1

Area of polygon 2

### NMTC 2019 Inter Stage 1 Question 18

In a rectangle , points are taken on the sides respectively such that the lengths and are integers and is rectangle. The largest possible area of PQRS is

Area of

### NMTC 2019 Inter Stage 1 Question 21

is a point inside an equilateral triangle . The perpendicular distance to the sides of the triangle are in the ratio . If where are co-prime positive integers, then equals

Let

Similarly

Area of

### NMTC 2019 Inter Stage 1 Question 22

In and . Point lies on and bisects . Point lies on and bisects . If the bisectors intersect at , then the ratio

bisect

Length of
so
and also bisect

Ans.

### NMTC 2019 Inter Stage 1 Question 24

In quadrilateral and . If is an integer then

Difference of two side is less than third side

in

also in
so

### NMTC 2019 Primary Stage 1 Question 15

A box of dimension units is used to keep smaller cuboidal boxes so that no space is left between the boxes. If the box is packed with 100 such smaller boxes of the same size, then dimension of the smaller box is

Dimension of bigger box
No. of smaller box

Volume of smaller box

### NMTC 2019 Primary Stage 1 Question 21

Small rectangular sheets of length units and breadth units are available. These sheets are assembled and pasted in a big cardboard sheet, edge to edge and made into a square. The minimum number of such sheet required is

Area of square
No. of rectangle

Therefore

### NMTC 2019 Sub junior Stage 1 Question 7

is a square. is one fourth of the way from to and is one fourth of the way from to C. is the centre of the square. Side of the square is . Then the area of the shaded region in the figure in is

Draw perpendicular from to \& on .

So required area Area of Area of Area of quadrilateral .

Therefore

### NMTC 2019 Sub junior Stage 1 Question 8

is a rectangle with and are midpoints of and respectively and is the mid-point of . The ratio of the area of to area of is

Let length of rectangle
So area of trapezium

### NMTC 2019 Sub junior Stage 1 Question 17

In quadrilateral and . If is an integer then

Extend to such that

So by SAS rule

CPCT

So of of

So (1)
In

### NMTC 2019 Sub junior Stage 1 Question 19

In the given figure, is a right angled triangle with are points on , respectively such that and . Then, (in degree).

Let

So

So

### NMTC 2019 Sub Junior Stage 1 Question 21

The area of a sector and the length of the arc of the sector are equal in numerical value. Then the radius of the circle is

Area of sector = length of are of sector

### NMTC 2019 Junior Stage 1 Question 2

In , the medians through and are perpendicular. Then is equal to

Let

In

In

In

Equation (i) + (ii)

Option (D).

### NMTC 2019 Junior Stage 1 Question 3

(B) ABCD has an in-circle
(C) ABCD has both circum-circle and in-circle
(D) It has neither a circum-circle nor an in-circle

For incircle

In circle is possible
for cyclic quadrilateral (circumcircle) theorem should be followed.

It is not a cyclic quadrilateral

Option B is correct

### NMTC 2019 Junior Stage 1 Question 5

In a rhombus of side length 5 , the length of one of the diagonals is at least 6 , and the length of the other diagonal is at most 6 . What is the maximum value of the sum of the diagonals ?
(A)
(B) 14
(C)
(D) 12

Let diagonal are and

We have to find ?

By option

from here we get it is not possible.

By option (B)

it is possible maximum value which is greater by other two options.

### NMTC 2019 Junior Stage 1 Question 9

The number of acute angled triangles whose vertices are chosen from the vertices of a rectangular box is
(A) 6
(B) 8
(C) 12
(D) 24

From each surface diagonals there are 4 triangles are possible which are equilateral of side
each.

But in this manner each triangle is counted thrice therefore
Option (B).

### NMTC 2019 Junior Stage 1 Question 21

Circles A, B and C are externally tangent to each other and internally tangent to circle D. Circles A and are congruent. Circle has radius 1 unit and passes through the centre of circle D. Then the radius of circle is units.

In

In CAN

### NMTC 2019 Junior Stage 1 Question 24

In shows below, is a point on and a point on such that is a point in the interior of is a point on and is a point on such that = BG. Also, . The measure of in degree is

As

Therefore

### NMTC 2019 Stage 1 Inter Question 5

The area of the curve enclosed by is :

(A) 16
(B) 12
(C) 8
(D) 4

Area of is equivalent to area of

So it is a rhombus

Hence area will be,

### NMTC 2019 Inter Stage 1 Question 11

In a rectangle , point lies on such that and point lies on such that 2. Lines and intersect at and respectively. If , are relatively prime positive integers, then the minimum value of is :

(i)
.(ii)

### NMTC 2019 Inter Stage 1 Question 13

A regular polygon has 100 sides each of length. A another regular polygon has 200 sided each of length 2. When the area of the larger polygon is divided by the area of the smaller polygon, the quotient is closest to the integer
(A) 2
(B) 4
(C) 8
(D) 16

Area of polygon 1

Area of polygon 2

### NMTC 2019 Inter Stage 1 Question 18

In a rectangle , points are taken on the sides respectively such that the lengths and are integers and is rectangle. The largest possible area of PQRS is

Area of

### NMTC 2019 Inter Stage 1 Question 21

is a point inside an equilateral triangle . The perpendicular distance to the sides of the triangle are in the ratio . If where are co-prime positive integers, then equals

Let

Similarly

Area of

### NMTC 2019 Inter Stage 1 Question 22

In and . Point lies on and bisects . Point lies on and bisects . If the bisectors intersect at , then the ratio

bisect

Length of
so
and also bisect

Ans.

### NMTC 2019 Inter Stage 1 Question 24

In quadrilateral and . If is an integer then

Difference of two side is less than third side

in

also in
so

### NMTC 2019 Primary Stage 1 Question 15

A box of dimension units is used to keep smaller cuboidal boxes so that no space is left between the boxes. If the box is packed with 100 such smaller boxes of the same size, then dimension of the smaller box is

Dimension of bigger box
No. of smaller box

Volume of smaller box

### NMTC 2019 Primary Stage 1 Question 21

Small rectangular sheets of length units and breadth units are available. These sheets are assembled and pasted in a big cardboard sheet, edge to edge and made into a square. The minimum number of such sheet required is

Area of square
No. of rectangle

Therefore

### NMTC 2019 Sub junior Stage 1 Question 7

is a square. is one fourth of the way from to and is one fourth of the way from to C. is the centre of the square. Side of the square is . Then the area of the shaded region in the figure in is

Draw perpendicular from to \& on .

So required area Area of Area of Area of quadrilateral .

Therefore

### NMTC 2019 Sub junior Stage 1 Question 8

is a rectangle with and are midpoints of and respectively and is the mid-point of . The ratio of the area of to area of is

Let length of rectangle
So area of trapezium

### NMTC 2019 Sub junior Stage 1 Question 17

In quadrilateral and . If is an integer then

Extend to such that

So by SAS rule

CPCT

So of of

So (1)
In

### NMTC 2019 Sub junior Stage 1 Question 19

In the given figure, is a right angled triangle with are points on , respectively such that and . Then, (in degree).

Let

So

So

### NMTC 2019 Sub Junior Stage 1 Question 21

The area of a sector and the length of the arc of the sector are equal in numerical value. Then the radius of the circle is

Area of sector = length of are of sector

### NMTC 2019 Junior Stage 1 Question 2

In , the medians through and are perpendicular. Then is equal to

Let

In

In

In

Equation (i) + (ii)

Option (D).

### NMTC 2019 Junior Stage 1 Question 3

(B) ABCD has an in-circle
(C) ABCD has both circum-circle and in-circle
(D) It has neither a circum-circle nor an in-circle

For incircle

In circle is possible
for cyclic quadrilateral (circumcircle) theorem should be followed.

It is not a cyclic quadrilateral

Option B is correct

### NMTC 2019 Junior Stage 1 Question 5

In a rhombus of side length 5 , the length of one of the diagonals is at least 6 , and the length of the other diagonal is at most 6 . What is the maximum value of the sum of the diagonals ?
(A)
(B) 14
(C)
(D) 12

Let diagonal are and

We have to find ?

By option

from here we get it is not possible.

By option (B)

it is possible maximum value which is greater by other two options.

### NMTC 2019 Junior Stage 1 Question 9

The number of acute angled triangles whose vertices are chosen from the vertices of a rectangular box is
(A) 6
(B) 8
(C) 12
(D) 24

From each surface diagonals there are 4 triangles are possible which are equilateral of side
each.

But in this manner each triangle is counted thrice therefore
Option (B).

### NMTC 2019 Junior Stage 1 Question 21

Circles A, B and C are externally tangent to each other and internally tangent to circle D. Circles A and are congruent. Circle has radius 1 unit and passes through the centre of circle D. Then the radius of circle is units.

In

In CAN

### NMTC 2019 Junior Stage 1 Question 24

In shows below, is a point on and a point on such that is a point in the interior of is a point on and is a point on such that = BG. Also, . The measure of in degree is

As

Therefore

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