NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and its Properties

NCERT Solutions for Maths Chapter 6 The Triangle and Its Properties class 7 Topic 6.1

Q: 1 Write the six elements (i.e., the 3 sides and the 3 angles) of .

Answer: The Triangle .

The Elements of the triangle are:

Sides :

Angles :

2. Write the:

(i) Side opposite to the vertex Q of

(ii) Angle opposite to the side LM of

(iii) Vertex opposite to the side RT of

Answer: (i) The side opposite to the vertex Q of

(ii) Angle opposite to the side LM of

(iii) Vertex opposite to the side RT of = Vertex S.

3. Look at Fig and classify each of the triangles according to its

(a) SidesThe lengths of two sides of a triangle are 12 cm and 15 cm. Between what two measures should the length of the third side fall?

(b) Angles

Answer: i) The triangle ABC

Based on Side: In Triangle ABC, Since two sides (BC and AC ) are equal (= 8 cm ) The given triangle is an isosceles triangle.

Based on Angle: In Triangle ABC, Since all the triangles are less than 90 degrees, So the given triangle is Acute angled triangle.

ii) The Triangle PQR

Based on Side : In Triangle PQR, All the sides are different so, The given triangle is a scalene triangle .

Based on Angle: In Triangle PQR, Since angle QRP is a right angle, So the given triangle is Right-angled triangle.

iii) The Triangle LMN

Based on Side : In Triangle LMN, Since two sides (MN and NL ) are equal (=7 cm ), The given triangle is an isosceles triangle.

Based on Angle: In Triangle LMN, Since angle MNL is an Obtuse angle, So the given triangle is Obtuse angled triangle.

iv) The Triangle RST

Based on Side : In Triangle RST, All the sides are equal (=5.2 cm) so, The given triangle is an Equilateral triangle.

Based on Angle: In Triangle RST, Since all the triangles are less than 90 degrees, So the given triangle is Acute angled triangle.

v) The triangle ABC

Based on Side : In Triangle ABC, Since two sides (AB and BC ) are equal (= 3 cm ) The given triangle is an isosceles triangle.

Based on Angle: In Triangle ABC, Since angle ABC is greater than 90 degrees So the given triangle is Obtuse angled triangle.

vi) The Triangle PQR

Based on Side : In Triangle PQR, Since two sides (PQ and QR ) are equal (= 6 cm ) The given triangle is an isosceles triangle .

Based on Angle: In Triangle PQR, Since angle PQR is a right angle, So the given triangle is Right-angled triangle.

NCERT Solutions for Maths Chapter 6 The Triangle and Its Properties class 7 Topic 6.2

1. How many medians can a triangle have?

Answer: In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side. Every triangle has exactly three medians, one from each vertex.

2. Does a median lie wholly in the interior of the triangle? (If you think that this is not true, draw a figure to show such a case).

Answer: Yes, The median always lies in the interior of the triangle.

As we can see in all three cases the median lies inside the triangle.

NCERT Solutions for Maths Chapter 6 The Triangle and Its Properties class 7 Topic 6.3

1. How many altitudes can a triangle have?

Answer: Every triangle has three bases (any of its sides) and three altitudes (heights). Every altitude is the perpendicular segment from a vertex to its opposite side (or the extension of the opposite side.

2. Draw rough sketches of altitudes from A to for the following triangles (Fig ):

Answer: altitudes from A to for the triangles are:

3. Will an altitude always lie in the interior of a triangle? If you think that this need not be true, draw a rough sketch to show such a case.

Answer: No, the altitude of a triangle might lie outside the triangle. for example in the obtuse-angled triangle, we have to extend the base side for making altitude angle.

Altitudes

4. Can you think of a triangle in which two altitudes of the triangle are two of its sides?

Answer: Yes, in a Right-angled triangle, the two altitudes of the sides are two sides as they make an angle of 90 degrees with one another.

5. Can the altitude and median be same for a triangle?

( Hint : For Q.No. 4 and 5, investigate by drawing the altitudes for every type of triangle).

Answer: Yes, the altitude and median can be the same in a triangle. for example, consider an equilateral triangle, the median which divides the side in equal is also perpendicular to the side and hence the altitude and the median is the same.

NCERT solutions for maths chapter 6 the triangle and its properties class 7 topic 6.4

1. Exterior angles can be formed for a triangle in many ways. Three of them are shown here (Fig )

There are three more ways of getting exterior angles. Try to produce those rough sketches.

Answer: In a triangle, there are a total of six exterior angles

Exterior Angles in a Triangle

2. Are the exterior angles formed at each vertex of a triangle equal?

Answer: No, The exterior angle formed at the vertices of a triangle are not equal. The exterior angle is equal to the sum of the two opposite interior angles.

3. What can you say about the sum of an exterior angle of a triangle and its adjacent interior angle?

Answer: As we can see they both angles forms a straight line, Hence the sum of an exterior angle of a triangle and its adjacent interior angle is always .

1. What can you say about each of the interior opposite angles, when the exterior angle is

(i) a right angle? (ii) an obtuse angle? (iii) an acute angle?

Answer: (i) a right angle

When the exterior angle is , the sum of opposite internal angles is

(ii) an obtuse angle

When the exterior angle is the obtuse angle, the opposite interior angles can be both acute, one right and one acute and one obtuse and one acute.

(iii) an acute angle?

When the exterior angle is an acute angle, both the internal angle has to be acute angles.

2. Can the exterior angle of a triangle be a straight angle?

Answer: No, the exterior angle of a triangle cannot be a straight angle because if the exterior angle is straight then there won't be any triangle left that would be a straight line. (imagine it visually).

1. An exterior angle of a triangle is of measure and one of its interior opposite angles is of measure . Find the measure of the other interior opposite angle.

Answer: As we know that in a triangle, the exterior angle is equal to the sum of opposite interior angles.

So,

According to the question,

Hence the other interior angle is .

2. The two interior opposite angles of an exterior angle of a triangle are and . Find the measure of the exterior angle.

Answer: As we know that in a triangle, the exterior angle is equal to the sum of opposite interior angles.

So,

Exterior Angle = + .

=

Hence the measure of the exterior is .

3. Is something wrong in this diagram (Fig )? Comment.

Answer: Yes, The measure of the exterior angle is given wrong.

As we know that in a triangle, the exterior angle is equal to the sum of opposite interior angles.

So,

Exterior angle =

=

Hence the exterior angle should be equal be instead of .

NCERT Solutions for Maths Chapter 6 The Triangle and Its Properties class 7 Topic 6.5

1. Two angles of a triangle are and . Find the third angle.

Answer: Let the third angle be

Now, As we know the sum of internal angles of a triangle is 180. so,

Hence the Third angle is .

2. One of the angles of a triangle is and the other two angles are equal. Find the measure of each of the equal angles.

Answer: Let the two same angles in triangle be .

Now, As we know the sum of internal angles of a triangle is 180. so,

.

Hence both other angles are each.

3. The three angles of a triangle are in the ratio . Find all the angles of the triangle. Classify the triangle in two different ways.

Answer: Let the angles of the triangles be

So,

As we know the sum of internal angles of a triangle is 180. so,

Hence the angles of the triangles are .

On the Basis of sides, the triangle is isosceles triangle as two sides of the triangle are equal.

On the Basis of angle, the triangle is Right-Angled Triangle as it has one angle equal to 90 degrees.

1. Can you have a triangle with two right angles?

Answer: No, we cannot have a triangle with two right angles. as the sum of the angles of a triangle is always 180 degrees, If there are two right angles then the sum would exceed 180 which is not possible.

2. Can you have a triangle with two obtuse angles?

Answer: No, we can not have two obtuse angles in a triangle because the sum of angles of the triangle is always 180 degrees and if there are two obtuse angles in the triangle then the sum would be more than 180 degrees which are not possible.

3. Can you have a triangle with two acute angles?

Answer: Yes, of course! we can have a triangle with two acute angles. all the obtuse-angled triangles have two acute angles in them.

4. Can you have a triangle with all the three angles greater than ?

Answer: No, we can not have a triangle with all the three angles greater than because then the sum of the angles of the triangles would be greater than 180 degree which is no possible.

5. Can you have a triangle with all the three angles equal to ?

Answer: Yes, we can have a triangle with all the three angles equal to as the sum of the angles of the triangle would be exactly 180 degrees. such triangles are called equilateral triangles.

6. Can you have a triangle with all the three angles less than ?

Answer: No, we can not have a triangle with all angles less than because then the sum of the angles would be less than 180 degrees which are also not possible.

The sum of angles of a triangle is exactly neither more than that nor less than that.

NCERT Solutions for Maths Chapter 6 The Triangle and Its Properties class 7 Topic 6.6

1. Find angle x in each figure:

Answer: As we know, in an isosceles triangle, two sides and the angles they make with the third side are equal.

and

the sum of angles of the triangle is equal to . So,

i)

ii)

iii)

iv)

v)

vi)

vii)

viii) As we know, the exterior angle is equal to the sum of opposite internal angles in a triangle. So,

ix) As we know when two lines are intersecting, the opposite angles are equal. So

2. Find angles x and y in each figure.

Answer: i) As we know, in an isosceles triangle, two sides and the angles they make with the third side are equal. So,

Now, As we know the sum of internal angles of a triangle is 180. so,

Hence, .

ii) As we know, in an isosceles triangle, two sides and the angles they make with the third side are equal.

AND

the sum of internal angles of a triangle is 180. so,

Also,

.

Hence .

iii) As we know when two lines are intersecting, the opposite angles are equal.

And

the sum of internal angles of a triangle is 180. so,

Now, As we know, the exterior angle is equal to the sum of opposite internal angles in a triangle

Hence .

NCERT Solutions for Maths Chapter 6 The Triangle and Its Properties class 7 Topic 6.7

1. Is the sum of any two angles of a triangle always greater than the third angle?

Answer: No, we can not say that. because we can have triangles in which the sum of two angles is less than the third angle. For Example:

A triangle ABC with

NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and Its Properties Topic 6.8

Q. Find the unknown length x in the following figures (Fig ):

Answer: As we know in a Right-angled Triangle: By Pythagoras Theorem,

So, using this theorem,

i)

.

ii)

iii)

iv)

v) in this question as we can see from the figure, it is making the right angle with the half-length of x, so

.

vi)

1. Which is the longest side in the triangle PQR, right-angled at P?

Answer: The hypotenuse is the longest side in a triangle. So when the right angle is at P, the Longest side would be QR.

2. Which is the longest side in the triangle ABC, right-angled at B?

Answer: The hypotenuse is the longest side in a triangle. So when the right angle is at B, the Longest side would be AC.

3. Which is the longest side of a right triangle?

Answer: The hypotenuse is the longest side in a Right-angled triangle.

4. ‘The diagonal of a rectangle produce by itself the same area as produced by its length and breadth’– This is Baudhayan Theorem. Compare it with the Pythagoras property.

Answer: Baudhayan Theorem and Pythagoras theorem are basically the same. Baudhayan Theorem used geometry to intuitively prove the Pythagoras theorem.

NCERT Solutions for Class 7 Triangles and Its Properties Maths Chapter 6 Exercise 6.1

1. In , D is the mid-point of .

is _________________.

PD is _________________.

Is ?

Answer: is the Altitude of the triangle.

PD is the Mediun of the triangle.

No, . As QD = DR.

2. Draw rough sketches for the following:

(a) In , BE is a median.

(b) In , PQ and PR are altitudes of the triangle.

(c) In , YL is an altitude in the exterior of the triangle.

Answer: (a) In , BE is a median.

(b) In , PQ and PR are altitudes of the triangle.

(c) In , YL is an altitude in the exterior of the triangle.

3. Verify by drawing a diagram if the median and altitude of an isosceles triangle can be same.

Answer: Yes, it is very much possible that the median and altitude of an isosceles triangle is the same. for example, The given triangle has the same median and altitude.

NCERT Solutions for Class 7 Triangles and Its Properties Maths Chapter 6 Exercise 6.2

1. Find the value of the unknown exterior angle x in the following diagrams:

Answer: As we know that the exterior angle is equal to the sum of the opposite internal angles. So,

i)

ii)

iii)

iv)

v)

vi)

2. Find the value of the unknown interior angle in the following figures:

Answer: As we know that the exterior angle is equal to the sum of the opposite internal angles. So,

i)

ii)

iii)

iv)

v)

vi)

NCERT Solutions for Class 7 Triangles and Its Properties Maths Chapter 6 Exercise 6.3

1. Find the value of the unknown x in the following diagrams

Answer: As we know that the sum of the internal angles of the triangle is equal to . So,

i)

ii)

iii)

iv)

v)

vi)

2. Find the values of the unknowns x and y in the following diagrams

Answer: i) As we know, the exterior angle is equal to the sum of opposite internal angles in a triangle.

Now, As we know the sum of internal angles of a triangle is 180. so,

Hence, .

ii) As we know when two lines are intersecting, the opposite angles are equal. So

Now, As we know the sum of internal angles of a triangle is 180. so,

Hence, .

iii) As we know, the exterior angle is equal to the sum of opposite internal angles in a triangle

Now, As we know the sum of internal angles of a triangle is 180. so,

Hence, .

iv)

As we know when two lines are intersecting, the opposite angles are equal. So

Now, As we know the sum of internal angles of a triangle is 180. so,

Hence,

v) As we know when two lines are intersecting, the opposite angles are equal. So

Now, As we know the sum of internal angles of a triangle is 180. so,

Hence,

vi) As we know when two lines are intersecting, the opposite angles are equal. So

Now, As we know the sum of internal angles of a triangle is 180. so,

Hence, .

NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and Its Properties Exercise 6.4

1. Is it possible to have a triangle with the following sides?

(i) 2 cm, 3 cm, 5 cm (ii) 3 cm, 6 cm, 7 cm
(iii) 6 cm, 3 cm, 2 cm

Answer: As we know, According to the Triangle inequality law, the sum of lengths of any two sides of a triangle would always greater than the length of the third side. So

Verifying this inequality by taking all possible combinations, we have,

(i) 2 cm, 3 cm, 5 cm

3 + 5 > 2 ----> True

2 + 5 > 3----> True

2 + 3 > 5 ---->False

Hence the triangle is not possible.

(ii) 3 cm, 6 cm, 7 cm
3 + 6 > 7 -----> True

3 + 7 > 6 ------>True

6 + 7 > 3 ------>True

Hence, The triangle is possible.

(iii) 6 cm, 3 cm, 2 cm

6 + 3 > 2 ------>True

6 + 2 > 3 ------> True

3 + 2 > 6 ------->False

Hence triangle is not possible.

2. Take any point O in the interior of a triangle PQR. Is

(i) ?
(ii) ?
(iii) ?

Answer: i) As POQ is a triangle, the sum of any two sides will always be greater than the third side. so

Yes, .

ii) As ROQ is a triangle, the sum of any two sides will always be greater than the third side. so

Yes,

iii) As ROQ is a triangle, the sum of any two sides will always be greater than the third side. so

Yes,

3. AM is a median of a triangle ABC.
Is ?
(Consider the sides of triangles and .)

Answer: As we know that the sum of two sides of ANY triangle is always greater than the third side(Triangles Inequality Rule).

So,

In :

In :

Adding (1) and (2), we get

As we can see M is the point in line BC So, we can say

So our equation becomes

.

Hence it is a True statement.

4. ABCD is a quadrilateral.
Is ?

Answer: As we know that the sum of two sides of ANY triangle is always greater than the third side(Triangles Inequality Rule).

So,

In :

In :

Adding (1) and (2) we get,

Hence the given statement is True.

5. ABCD is quadrilateral. Is
?

Answer: Let the intersection point of the two diagonals be O.

As we know that the sum of two sides of ANY triangle is always greater than the third side(Triangles Inequality Rule).

So,

In :

In :

In :

In :

Now, Adding all four equations we, get

which can also be expressed as

Hence this is true.

6. The lengths of two sides of a triangle are 12 cm and 15 cm. Between what two measures should the length of the third side fall?

Answer: Let ABC be a triangle with AB = 12cm and BC = 15cm

Now

As we know that the sum of two sides of ANY triangle is always greater than the third side(Triangles Inequality Rule).

AB + BC > CA

12 + 15 > CA

CA < 27......(1)

Also, in a similar way

AB + CA > BC

CA > BC - AB

CA > 15 - 12

CA > 3............(2)

Hence from (1) and (2), we can say that the length of third side of the triangle must be between 3cm to 27 cm.

NCERT Solutions for Class 7 Maths Chapter 6 The Triangle and Its Properties Exercise 6.5

1. PQR is a triangle, right-angled at P. If and , find QR.

Answer: As we know,

In a Right-angled Triangle: By Pythagoras Theorem,

As PQR is a right-angled triangle with

Base = PQ = 10 cm.

Perpendicular = PR = 24 cm.

Hypotenuse = QR

So, By Pythagoras theorem,

Hence, Length od QR is 26 cm.

2. ABC is a triangle, right-angled at C. If and , find BC.

Answer: As we know,

In a Right-angled Triangle: By Pythagoras Theorem,

As ABC is a right-angled triangle with

Base = AC= 7 cm.

Perpendicular = BC

Hypotenuse = AB = 25 cm

So, By Pythagoras theorem,

Hence, Length od BC is 24 cm.

3. A 15 m long ladder reached a window 12 m high from the ground on placing it against a wall at a distance a. Find the distance of the foot of the ladder from the wall.

Answer: Here. As we can see, The ladder with wall forms a right-angled triangle with

the vertical height of the wall = perpendicular = 12 m

length of ladder = Hypotenuse = 15 m

Now, As we know

In a Right-angled Triangle: By Pythagoras Theorem,

Hence the distance of the foot of the ladder from the wall is 9 m.

4. Which of the following can be the sides of a right triangle?

(i) , , 6 cm.

(ii) 2 cm, 2 cm, 5 cm.

(iii) , 2cm,

In the case of right-angled triangles, identify the right angles.

Answer: As we know,

In a Right-angled Triangle: By Pythagoras Theorem,

(i) , , 6 cm.

As we know the hypotenuse is the longest side of the triangle, So

Hypotenuse = 6.5 cm

Verifying the Pythagoras theorem,

Hence it is a right-angled triangle.

The Right-angle lies on the opposite of the longest side (hypotenuse) So the right angle is at the place where 2.5 cm side and 6 cm side meet.

(ii) 2 cm, 2 cm, 5 cm.

As we know the hypotenuse is the longest side of the triangle, So

Hypotenuse = 5 cm

Verifying the Pythagoras theorem,

Hence it is Not a right-angled triangle.

(iii) , 2cm, .

As we know the hypotenuse is the longest side of the triangle, So

Hypotenuse = 2.5 cm

Verifying the Pythagoras theorem,

Hence it is a Right-angled triangle.

The right angle is the point where the base and perpendicular meet.

5. A tree is broken at a height of 5 m from the ground and its top touches the ground at a distance of 12 m from the base of the tree. Find the original height of the tree.

Answer: As we can see the tree makes a right angle with

Perpendicular = 5 m

Base = 12 m

As we know,

In a Right-angled Triangle: By Pythagoras Theorem,

Here, The Hypotenuse of the triangle was also a part of the tree originally, So

The Original height of the tree = height + hypotenuse

= 5 m + 13 m

= 18 m.

Hence the original height of the tree was 18 m.

6. Angles Q and R of a are and . Write which of the following is true:

(i)

(ii)

(iii)

Answer: As we know the sum of the angles of any triangle is always 180. So,

Now. Since PQR is a right-angled triangle with right angle at P. So

Hence option (ii) is correct.

7. Find the perimeter of the rectangle whose length is 40 cm and a diagonal is 41 cm.

Answer: As we can see in the rectangle,

By Pythagoras theorem,

Now as given in the question,

Diagonal = 41 cm.

Length = 40 cm.

So, Putting these value we get,

Hence the width of the rectangle is 9 cm.

So

The perimeter of the rectangle = 2 ( Length + Width )

= 2 ( 40 cm + 9 cm )

= 2 x 49 cm

= 98 cm

Hence the perimeter of the rectangle is 98 cm.

8. The diagonals of a rhombus measure 16 cm and 30 cm. Find its perimeter.

Answer: As we know that the diagonals of the rhombus are perpendicular to each other and intersect at a point which is mid of both the diagonal.

So. By Pythagoras Theorem we can say that

Hence Side of the rhombus is 17 cm.

So,

The Perimeter of the rhombus = 4 x 17 cm

= 68 cm.

Hence, the perimeter of the rhombus is 68 cm.