NCERT solutions for class 7 maths chapter 6 The Triangle and its Properties

(i) Altitude

An altitude has one endpoint at a vertex of the triangle and another on the line containing the opposite side.

(ii) Median

A median connects a vertex of a triangle to the midpoint of the opposite side.

(iii) No, QM ≠ MR because D is the midpoint of QR.

 

Draw a line segment PS ⊥ BC. It is an altitude for this triangle. Here, we observe that the length of QS and SR is also the same. So PS is also a median of this triangle.

 

(a) A median connects a vertex of a triangle to the midpoint of the opposite side.

(b)

An altitude has one endpoint at a vertex of the triangle and another on the line containing the opposite side.

(c)

In the figure, we may observe that for ΔXYZ, YL is an altitude drawn exteriorly to side XZ which is extended up to point L.

 

(i) We know that,

An exterior angle of a triangle is equal to the sum of its interior opposite angles.

= x = 50o + 70o

= x = 120o

(ii) We know that,

An exterior angle of a triangle is equal to the sum of its interior opposite angles.

= x = 65o + 45o

= x = 110o

(iii) We know that,

An exterior angle of a triangle is equal to the sum of its interior opposite angles.

= x = 30o + 40o

= x = 70o

(iv) We know that,

An exterior angle of a triangle is equal to the sum of its interior opposite angles.

= x = 60o + 60o

= x = 120o

(v) We know that,

An exterior angle of a triangle is equal to the sum of its interior opposite angles.

= x = 50o + 50o

= x = 100o

(vi) We know that,

An exterior angle of a triangle is equal to the sum of its interior opposite angles.

= x = 30o + 60o

= x = 90o

 

 

(i) We know that,

An exterior angle of a triangle is equal to the sum of its interior opposite angles.

= x + 50o = 115o

By transposing 50o from LHS to RHS, it becomes – 50o

= x = 115o – 50o

= x = 65o

(ii) We know that,

An exterior angle of a triangle is equal to the sum of its interior opposite angles.

= 70o + x = 100o

By transposing 70o from LHS to RHS, it becomes – 70o

= x = 100o – 70o

= x = 30o

(iii) We know that,

An exterior angle of a triangle is equal to the sum of its interior opposite angles.

The given triangle is a right-angled triangle. So, the angle opposite to the x is 90o.

= x + 90o = 125o

By transposing 90o from LHS to RHS, it becomes – 90o

= x = 125o – 90o

= x = 35o

(iv) We know that,

An exterior angle of a triangle is equal to the sum of its interior opposite angles.

= x + 60o = 120o

By transposing 60o from LHS to RHS, it becomes – 60o

= x = 120o – 60o

= x = 60o

(v) We know that,

An exterior angle of a triangle is equal to the sum of its interior opposite angles.

The given triangle is a right-angled triangle. So, the angle opposite to the x is 90o.

= x + 30o = 80o

By transposing 30o from LHS to RHS, it becomes – 30o

= x = 80o – 30o

= x = 50o

(vi) We know that,

An exterior angle of a triangle is equal to the sum of its interior opposite angles.

The given triangle is a right-angled triangle. So, the angle opposite to the x is 90o.

= x + 35o = 75o

By transposing 35o from LHS to RHS, it becomes – 35o

= x = 75o – 35o

= x = 40o

 

 

(i)

We know that,

The sum of all the interior angles of a triangle is 180o.

Then,

= ∠BAC + ∠ABC + ∠BCA = 180o

= x + 50o + 60o = 180o

= x + 110o = 180o

By transposing 110o from LHS to RHS it becomes – 110o

= x = 180o – 110o

= x = 70o

(ii)

We know that,

The sum of all the interior angles of a triangle is 180o.

The given triangle is a right angled triangle. So the ∠QPR is 90o.

Then,

= ∠QPR + ∠PQR + ∠PRQ = 180o

= 90o + 30o + x = 180o

= 120o + x = 180o

By transposing 110o from LHS to RHS it becomes – 110o

= x = 180o – 120o

= x = 60o

(iii)

We know that,

The sum of all the interior angles of a triangle is 180o.

Then,

= ∠XYZ + ∠YXZ + ∠XZY = 180o

= 110o + 30o + x = 180o

= 140o + x = 180o

By transposing 140o from LHS to RHS it becomes – 140o

= x = 180o – 140o

= x = 40o

(iv)

We know that,

The sum of all the interior angles of a triangle is 180o.

Then,

= 50o + x + x = 180o

= 50o + 2x = 180o

By transposing 50o from LHS to RHS it becomes – 50o

= 2x = 180o – 50o

= 2x = 130o

= x = 130o/2

= x = 65o

(v)

We know that,

The sum of all the interior angles of a triangle is 180o.

Then,

= x + x + x = 180o

= 3x = 180o

= x = 180o/3

= x = 60o

∴The given triangle is an equiangular triangle.

(vi)

We know that,

The sum of all the interior angles of a triangle is 180o.

Then,

= 90o + 2x + x = 180o

= 90o + 3x = 180o

By transposing 90o from LHS to RHS it becomes – 90o

= 3x = 180o – 90o

= 3x = 90o

= x = 90o/3

= x = 30o

Then,

= 2x = 2 × 30o = 60o

 

(i)

We Know That,

An exterior angle of a triangle is equal to the sum of its interior opposite angles.

Then,

= 50o + x = 120o

By transposing 50o from LHS to RHS it becomes – 50o

= x = 120o – 50o

= x = 70o 

We also know that,

The sum of all the interior angles of a triangle is 180o.

Then,

= 50o + x + y = 180o

= 50o + 70o + y = 180o

= 120o + y = 180o

By transposing 120o from LHS to RHS it becomes – 120o

= y = 180o – 120o 

= y = 60o

(ii)

From the rule of vertically opposite angles,

= y = 80o

Then,

We know that,

The sum of all the interior angles of a triangle is 180o.

Then,

= 50o + 80o + x = 180o

= 130o + x = 180o

By transposing 130o from LHS to RHS it becomes – 130o

= x = 180o – 130o 

= x = 50o

(iii)

We know that,

The sum of all the interior angles of a triangle is 180o.

Then,

= 50o + 60o + y = 180o

= 110o + y = 180o

By transposing 110o from LHS to RHS it becomes – 110o

= y = 180o – 110o 

= y = 70o

Now,

From the rule of linear pair,

= x + y = 180o

= x + 70o = 180o

By transposing 70o from LHS to RHS it becomes – 70o

= x = 180o – 70

= x = 110o

(iv)

From the rule of vertically opposite angles,

= x = 60o

Then,

We know that,

The sum of all the interior angles of a triangle is 180o.

Then,

= 30o + x + y = 180o

= 30o + 60o + y = 180o

= 90o + y = 180o

By transposing 90o from LHS to RHS it becomes – 90o

= y = 180o – 90o 

= y = 90o

(v)

From the rule of vertically opposite angles,

= y = 90o

Then,

We know that,

The sum of all the interior angles of a triangle is 180o.

Then,

= x + x + y = 180o

= 2x + 90o = 180o

By transposing 90o from LHS to RHS it becomes – 90o

= 2x = 180o – 90o 

= 2x = 90o

= x = 90o/2

= x = 45o

(vi)

From the rule of vertically opposite angles,

= x = y

Then,

We know that,

The sum of all the interior angles of a triangle is 180o.

Then,

= x + x + x = 180o

= 3x = 180o

= x = 180o/3

= x = 60o

 

Let us consider ABCD is quadrilateral and P is the point where the diagonals intersect. As shown in the figure below,

We know that,

The sum of the length of any two sides is always greater than the third side.

Now consider the ΔPAB,

Here, PA + PB > AB … [equation i]

Then, consider the ΔPBC

Here, PB + PC > BC … [equation ii]

Consider the ΔPCD

Here, PC + PD > CD … [equation iii]

Consider the ΔPDA

Here, PD + PA > DA … [equation iv]

By adding equation [i], [ii], [iii] and [iv] we get,

PA + PB + PB + PC + PC + PD + PD + PA > AB + BC + CD + DA

2PA + 2PB + 2PC + 2PD > AB + BC + CD + DA

2PA + 2PC + 2PB + 2PD > AB + BC + CD + DA

2(PA + PC) + 2(PB + PD) > AB + BC + CD + DA

From the figure we have, AC = PA + PC and BD = PB + PD

Then,

2AC + 2BD > AB + BC + CD + DA

2(AC + BD) > AB + BC + CD + DA

Hence, the given expression is true.

 

We know that,

The sum of the length of any two sides is always greater than the third side.

From the question, it is given that two sides of the triangle are 12 cm and 15 cm.

So, the third side’s length should be less than the sum of other two sides,

12 + 15 = 27 cm.

Then, it is given that the third side cannot not be less than the difference of the two sides, 15 – 12 = 3 cm

So, the length of the third side falls between 3 cm and 27 cm.

 

(i) Clearly, we have:

(2 + 3) = 5

5 = 5

Thus, the sum of any two of these numbers is not greater than the third.

Hence, it is not possible to draw a triangle whose sides are 2 cm, 3 cm and 5 cm.

(ii) Clearly, we have:

(3 + 6) = 9 > 7

(6 + 7) = 13 > 3

(7 + 3) = 10 > 6

Thus, the sum of any two of these numbers is greater than the third.

Hence, it is possible to draw a triangle whose sides are 3 cm, 6 cm and 7 cm.

(iii) Clearly, we have:

(3 + 2) = 5 < 6

Thus, the sum of any two of these numbers is less than the third.

Hence, it is not possible to draw a triangle whose sides are 6 cm, 3 cm and 2 cm.

 

We know that,

The sum of the length of any two sides is always greater than the third side.

Now consider the ΔABM,

Here, AB + BM > AM … [equation i]

Then, consider the ΔACM

Here, AC + CM > AM … [equation ii]

By adding equation [i] and [ii] we get,

AB + BM + AC + CM > AM + AM

From the figure we have, BC = BM + CM

AB + BC + AC > 2 AM

Hence, the given expression is true.

 

We know that,

The sum of the length of any two sides is always greater than the third side.

Now consider the ΔABC,

Here, AB + BC > CA … [equation i]

Then, consider the ΔBCD

Here, BC + CD > DB … [equation ii]

Consider the ΔCDA

Here, CD + DA > AC … [equation iii]

Consider the ΔDAB

Here, DA + AB > DB … [equation iv]

By adding equation [i], [ii], [iii] and [iv] we get,

AB + BC + BC + CD + CD + DA + DA + AB > CA + DB + AC + DB

2AB + 2BC + 2CD + 2DA > 2CA + 2DB

Take out 2 on both the side,

2(AB + BC + CA + DA) > 2(CA + DB)

AB + BC + CA + DA > CA + DB

Hence, the given expression is true.

 

If we take any point O in the interior of a triangle PQR and join OR, OP, OQ.

Then, we get three triangles ΔOPQ, ΔOQR and ΔORP is shown in the figure below.

We know that,

The sum of the length of any two sides is always greater than the third side.

(i) Yes, ΔOPQ has sides OP, OQ and PQ.

So, OP + OQ > PQ

(ii) Yes, ΔOQR has sides OR, OQ and QR.

So, OQ + OR > QR

(iii) Yes, ΔORP has sides OR, OP and PR.

So, OR + OP > RP

 

(i) Let a = 2.5 cm, b = 6.5 cm, c = 6 cm

Let us assume the largest value is the hypotenuse side i.e. b = 6.5 cm.

Then, by Pythagoras theorem,

b2 = a2 + c2

6.52 = 2.52 + 62

42.25 = 6.25 + 36

42.25 = 42.25

The sum of square of two side of triangle is equal to the square of third side,

∴The given triangle is right-angled triangle.

Right angle lies on the opposite of the greater side 6.5 cm.

(ii) Let a = 2 cm, b = 2 cm, c = 5 cm

Let us assume the largest value is the hypotenuse side i.e. c = 5 cm.

Then, by Pythagoras theorem,

c2 = a2 + b2

52 = 22 + 22

25 = 4 + 4

25 ≠ 8

The sum of square of two side of triangle is not equal to the square of third side,

∴The given triangle is not right-angled triangle.

(iii) Let a = 1.5 cm, b = 2 cm, c = 2.5 cm

Let us assume the largest value is the hypotenuse side i.e. b = 2.5 cm.

Then, by Pythagoras theorem,

b2 = a2 + c2

2.52 = 1.52 + 22

6.25 = 2.25 + 4

6.25 = 6.25

The sum of square of two side of triangle is equal to the square of third side,

∴The given triangle is right-angled triangle.

Right angle lies on the opposite of the greater side 2.5 cm.

 

Let ABC is the triangle and B is the point where tree is broken at the height 5 m from the ground.

Tree top touches the ground at a distance of AC = 12 m from the base of the tree,

By observing the figure we came to conclude that right angle triangle is formed at A.

From the rule of Pythagoras theorem,

BC2 = AB2 + AC2

BC2 = 52 + 122

BC2 = 25 + 144

BC2 = 169

BC = √169

BC = 13 m

Then, the original height of the tree = AB + BC

= 5 + 13

= 18 m

 

Given that ∠Q = 25o, ∠R = 65o

Then, ∠P =?

We know that sum of the three interior angles of triangle is equal to 180o.

∠PQR + ∠QRP + ∠RPQ = 180o

25o + 65o + ∠RPQ = 180o

90o + ∠RPQ = 180o

∠RPQ = 180 – 90

∠RPQ = 90o

Also, we know that side opposite to the right angle is the hypotenuse.

∴ QR2 = PQ2 + PR2

Hence, (ii) is true

 

 

Let ABCD be the rectangular plot.

Then, AB = 40 cm and AC = 41 cm

BC =?

According to Pythagoras theorem,

From right angle triangle ABC, we have:

= AC2 = AB2 + BC2

= 412 = 402 + BC2

= BC2 = 412 – 402

= BC2 = 1681 – 1600

= BC2 = 81

= BC = √81

= BC = 9 cm

Hence, the perimeter of the rectangle plot = 2 (length + breadth)

Where, length = 40 cm, breadth = 9 cm

Then,

= 2(40 + 9)

= 2 × 49

= 98 cm

 

 

Let PQRS be a rhombus, all sides of rhombus has equal length and its diagonal PR and SQ are intersecting each other at a point O. Diagonals in rhombus bisect each other at 90o.

So, PO = (PR/2)

= 16/2

= 8 cm

And, SO = (SQ/2)

= 30/2

= 15 cm

Then, consider the triangle POS and apply the Pythagoras theorem,

PS2 = PO2 + SO2

PS2 = 82 + 152

PS2 = 64 + 225

PS2 = 289

PS = √289

PS = 17 cm

Hence, the length of side of rhombus is 17 cm

Now,

Perimeter of rhombus = 4 × side of the rhombus

= 4 × 17

= 68 cm

∴ Perimeter of rhombus is 68 cm.

Let us draw a rough sketch of right-angled triangle

By the rule of Pythagoras Theorem,

Pythagoras theorem states that for any right angled triangle, the area of the square on the hypotenuse is equal to the sum of the areas of square on the legs.

In the above figure RQ is the hypotenuse,

QR2 = PQ2 + PR2

QR2 = 102 + 242

QR2 = 100 + 576

QR2 = 676

QR = √676

QR = 26 cm

Hence, the length of the hypotenuse QR = 26 cm.

 

Let us draw a rough sketch of right-angled triangle

By the rule of Pythagoras Theorem,

Pythagoras theorem states that for any right angled triangle, the area of the square on the hypotenuse is equal to the sum of the areas of square on the legs.

In the above figure RQ is the hypotenuse,

AB2 = AC2 + BC2

252 = 72 + BC2

625 = 49 + BC2

By transposing 49 from RHS to LHS it becomes – 49

BC2 = 625 – 49

BC2 = 576

BC = √576

BC = 24 cm

Hence, the length of the BC = 24 cm.

 

By the rule of Pythagoras Theorem,

Pythagoras theorem states that for any right angled triangle, the area of the square on the hypotenuse is equal to the sum of the areas of square on the legs.

In the above figure RQ is the hypotenuse,

152 = 122 + a2

225 = 144 + a2

By transposing 144 from RHS to LHS it becomes – 144

a2 = 225 – 144

a2 = 81

a = √81

a = 9 m

Hence, the length of a = 9 m.