Maths Chapter-06: The Triangle and its Properties

Exercise 6.1

Question 1:

In ? ABC, X is the midpoint of BC.

AP is _____________

AX is _____________

Is BP = PC?

Answer:

Given:

BX = XC

Therefore, AP is altitude.

AX is a median.

No, BP ? PC as X is the midpoint of BC

Question 2:

Draw a sketch for the following:

(i) In ? PQR, QX is a median

(ii) In ? ABC, AB and AC are altitudes of a triangle.

(iii) In ABC, BP is an altitude in the exterior of a triangle.

Answer:

(i) Here, QX is a median in ?PQR and PX = XR

(ii) Here, AB and AC are the altitudes of the ?ABC and CA ^ BA

(iii) BP is an altitude in the exterior of ?ABC

Question 3:

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

Answer:

Isosceles triangle means any two sides are same.

Take ?PQR and draw the median when PQ= PR

PX is the median and altitude of the given triangle.

Exercise 6.2

Question 1:

Find the value of the unknown exterior angle a in the following diagrams:

Answers:

Since the exterior angle = sum of the interior opposite angles, then

(a) a = 50⁰ + 70 ⁰= 120⁰

(b) a = 65⁰ + 45⁰ = 110⁰

( c) a = 30⁰ + 40⁰ = 70⁰

(d ) a = 60⁰ + 60⁰= 120⁰

(E) a = 50⁰ + 50⁰= 100⁰

(F) a = 60⁰+ 30⁰= 90⁰

Question 2:

Find the value of the unknown interior angle a in the following figures:

Answers:

Since the exterior angle = sum of the interior opposite angles, then

(a) a + 50^o = 115^o => a = 115^o – 50 ^o = 65 ^o

(b) 70 ^o + a = 100^o => a = 100^o – 70^o = 30^o

( c) a + 90^o = 125^o => a = 120 ^o – 90 ^o = 35 ^o

(d) 60 ^o + a = 120 ^o => a = 120 ^o – 60 ^o = 60 ^o

(e) 30 ^o + a = 80 ^o => a = 80 ^o +30 ^o = 50 ^o

(f) a + 35 ^o = 75 ^o => a = 75 ^o – 35 ^o = 40 ^o

Exercise 6.3

Question 1:

Find the value of the unknown a in the following given diagrams.

Answers:

(a) In \(\Delta\) PQR

\(\angle\) QPR + \(\angle\) PRQ + \(\angle\) PQR = 180^o [ By angle sum property of a triangle]

=> a + 50^o + 60^o = 180^o

=> a + 110^o = 180^o

=> a = 180^o – 110 = 70^o

(b) In \(\Delta\) ABC,

\(\angle\) CAB + \(\angle\) ABC + \(\angle\) CAB = 90^o [ By angle sum property of a triangle]

=> 90^o + 30^o + a = 180^o

=> a + 120^o= 180^o

=> a = 180^o – 120^o = 60^o

(c ) In \(\Delta\) ABC ,

\(\angle\) CAB + \(\angle\) ABC + \(\angle\) BCA = 180^o [ By angle of sum property of a triangle]

=> 30^o + 110^o+ a = 180^o

=> a + 140^o = 180^o

=> a = 180^o– 140^O = 40^O

(d) In the given isosceles triangle,

a + a + 50^o = 180^o

=> 2a + 50^o = 180^o

=> 2a = 180^o – 50^o

=> 2a = 130^o

=> \(a = \frac{130^{\circ}}{2}\) = 65^o

(e) In the given equilateral triangle,

a + a + a = 180^o[ By angle sum property of a triangle]

=> 3 a = 180^o

=> \(a = \frac{180^{\circ}}{3}\) = 60^o

(f) In the given right angled triangle,

a + 2a + 90^o = 180^o [ By the angle sum property of a triangle]

=> 3a + 90^o = 180^o

=> 3a = 180^o – 90^o

=> 3a = 90^o

=> \(a = \frac{90^{\circ}}{3}\) = 30^o

Question 2

Find the values a and b in the following diagrams that are given below:

Answers:

(a) 50^o+ a = 120^o [Exterior angle property of a \(\Delta\) ]

=> a = 120 ^o – 50 ^o = 70 ^o

Now, 50 ^o + a + b = 180 ^o [Angle sum property of a \(\Delta\) ]

=> 50 ^o + 70 ^o + b = 180 ^o

=> 120 ^o + b = 180 ^o

=> b = 180 ^o – 120 ^o = 60 ^o

(b) b = 80 ^o – – – – – – – – – – – (1) [ Vertically opposite angle]

Now, 50 ^o + a + b = 180 ^o [ Angle sum property of a \(\Delta\) ]

=> 50 ^o + 80 ^o + b = 180 ^o [From equation (1) ]

=> 130 ^o + b = 180 ^o

=> b = 180 ^o – 130 ^o = 50 ^o

( c) 50 ^o + 60 ^o = a [Exterior angle property of a \(\Delta\) ]

=> a = 110 ^o

Now, 50 ^o + 60 ^o + b = 180 ^o [Angle sum property of a \(\Delta\) ]

=> 110 ^o + b = 180 ^o

=> b = 180 ^o – 110 ^o

=> b = 70 ^o

(d) a = 60 ^o – – – – – – – – – – – – – – – (1) [ Vertically opposite angle]

Now, 30 ^o + a + b = 180 ^o [ Angle sum property of a \(\Delta\) ]

=> 50 ^o + 60 ^o + b = 180 ^o [From equation (1) ]

=> 90 ^o + b = 180 ^o

=> b = 180 ^o – 90 ^o = 90 ^o

(e) b = 90 ^o – – – – – – – – – – – – – (1) [ Vertically opposite angle]

Now, b + a + a = 180 ^o [ Angle sum property of a \(\Delta\) ]

=> 90 ^o + 2a = 180 ^o [From equation (1) ]

=> 2a = 180 ^o – 90 ^o

=> 2a = 90 ^o

=> \(a=\frac{90^{\circ}}{2}\) = 45^o

(f) a = b – – – – – – – – – – – (1) [ Vertically opposite angle]

Now, a + a + b = 180 ^o [ Angle sum property of a \(\Delta\) ]

=> 2a + a = 180 ^o [From equation (1) ]

=> 3a = 180 ^o

=> \(a=\frac{180^{\circ}}{3}\) = 60^o

Exercise 6.4

Question 1:

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

(a) 2cm, 3cm 5cm

(b) 3cm, 6cm, 7cm

(c) 6cm, 3cm, 2cm

Answer:

Since the triangle is possible having its sum of the lengths of any two sides would be greater than that length of the third side.

(a) 2cm , 3cm 5cm

2 + 3 > 5 No

2 + 5 > 3 Yes

3 + 5 > 2 Yes

This triangle is not possible.

(b) 3cm, 6cm and 7cm

3 + 6> 7 Yes

6 + 7 > 3 Yes

3 + 7 > 6 Yes

This triangle is possible.

( c) 6cm, 3cm and 2cm

6 + 3 > 2 Yes

6 + 2 > 3 Yes

2 + 3 > 6 No

This triangle is not possible.

Question 2:

Take any point P in the interior of a triangle ABC is:

(a) PA + PB > AB?

(b) PB + PC > BC?

(C ) PC + PA > CA?

Answers:

Join PC, PB and PA

(a) Is PA + PB > AB?

Yes, APB forms a triangle.

(b) Is PB + PC > BC?

Yes, CBP forms a triangle.

( c) Is PC + PA > CA?

Yes, CPA forms a triangle.

Question 3:

PO is a median of a triangle PQR. Is PQ + QR + RP > 2PO? (Consider the sides of a triangle as \(\Delta\) PQO and \(\Delta\) POR )

Answer:

Since, the sum of the lengths of any two sides in a triangle should be greater than the length of the third side.

Therefore,

In \(\Delta\) PQO, PQ + QO > PO – – – – – – – – – – (1)

In \(\Delta\) POR, PR + OQ > PO – – – – – – – – – – – (2)

Adding equation (1) and (2) we get,

PQ + PO + PR + OR > PO + PO

=> PQ + PR + ( QO + OR ) > 2PO

=> PQ + PR + QR > 2 PO

Hence, it is true.

Question 4:

PQRS is a quadrilateral. Is PQ + QR + RS + RP > PR + QS ?

Answer:

Since, the sum of lengths of any two sides in a triangle should be greater than the length of the third side.

Therefore,

In \(\Delta\) PQR, PQ + QR > PR – – – – – – – – – – (1)

In \(\Delta\) PSR, PS + SR > PR – – – – – – – – – (2)

In \(\Delta\) SRQ, SR + RQ > SQ – – – – – – – – – (3)

In \(\Delta\) PSQ, PS + PQ > SQ – – – – – – – – – – (4)

Adding the equations (1) , (2) , (3) and (4) we get,

PQ + QR + PS + SR + RQ + PS + PQ > PS + PS + SQ + SQ

=>( PQ + PQ ) + ( QR + QR ) + ( PS + PS ) + ( SR + SR ) > 2 PR + 2 SQ

=> 2 PQ + 2 QR + 2 PS + 2 SR > 2 PR + 2 SQ

=> 2 ( PQ + QR + PS + SR) > 2 ( PR + SQ )

=> PQ + QR + PS + SR > PR + SQ

=> PQ + QR + RS + SP > PR + SQ

Hence, it is true.

Question 5:

PQRS is a quadrilateral. Is PQ + QR + RS + SA < 2 (PR + QS)?

Answer:

Since the sum of the lengths of any two sides of a triangle should be greater than the length of the third side.

Therefore,

In \(\Delta\) PXQ, PQ < XP + XQ – – – – – – – – – – (1)

In \(\Delta\) QXR, QR < XQ + XR – – – – – – – – – – – (2)

In \(\Delta\) RXS, RS < XR + XS – – – – – – – – – – – (3)

In \(\Delta\) PXS, SP < XS + XP – – – – – – – – – – (4)

Adding the equations (1) , (2) , (3) and (4) We get,

PQ + QR + RS + SP < XP + XQ + XQ + XR + XR + XS + XS + XP

=> PQ + QR + RS + SP < 2XP + 2 XQ + 2 XR + XS

=> PQ + QR + RS + SP < 2[(PX + XR) + (XS + XQ)]

=> PQ + QR + RS + SP < 2[PR + QS]

Hence it is proved.

Question 6:

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

Answer:

Since, the sum of the lengths of any two sides of a triangle should be greater than the length of the third side.

It is given that the two sides of a triangle are 15cm and 12cm.

Therefore, the third side should be less than 15+ 12 = 27cm

Also that, the third side cannot be less than the difference of the other two sides.

Therefore, the third side has to be greater than 15-12 = 3cm.

Hence, the third side could be the length more than 3cm and less than 27cm.

Exercise. 6.5

Question 1:

ABC is a right angled triangle with \(\angle B = 90^{\circ}\). If AB = 12mm and BC = 5mm, find AC.

Answer:

Here AB = 12mm and BC = 5mm (given)

Assuming AC = y mm.

\(AC^{2} = AB^{2} + BC^{2}\)In \(\triangle ABC\) by applying Pythagoras theorem, we get

\( y^{2} = 12^{2} + 5^{2}\) \(\ therefore y^{2} = 144 + 25\) \(?y^{2} = 169\) \(? y = \sqrt{169} = 13mm\)

Thus, AC = 13mm

Question 2:

XYZ is a right angled triangle with \(\angle Y = 90^{\circ}\). If XZ = 5cm and YZ = 3cm, find XY.

Answer:

Assuming XY = acm.

Here XZ = 5cm and YZ = 3cm (given)

In \(\triangle XYZ\) by applying

Pythagoras theorem, we get

\(XZ^{2} = XY^{2} + YZ^{2}\) \(\ therefore a^{2} = 5^{2} – 3^{2}\) \(\ therefore a^{2} = 25 – 9 \) \(? a^{2} = 16\) \(? a = \sqrt{16} = 4cm\)

Thus, XY = 4cm

Question 3:

A 25cm long stick is rested on a wall at a point 24cm high on the wall. If the stick is at a distance x cm from wall on the ground, find the distance x.

Answer:

Let PR be the stick and P be the point where the stick touches the wall.

So, it becomes right angled triangle with \(\angle Q = 90^{\circ}\)

In \(\triangle PQR\) by applying Pythagoras theorem, we get

\( x^{2} =25^{2} – 24^{2}\)\(PR^{2} = PQ^{2} + QR^{2}\)

\(\ therefore x^{2} = 625 – 576 \) \(?x^{2} = 49\) \(? x = \sqrt{49} = 7cm\)

Thus, QR = 7cm

Question 4:

Which of the given below could be the lengths of the side of a right angled triangle?

5mm, 1.2mm, .9mm
3mm, 2mm, 1mm
1mm, 4mm, 9mm

Answer:

Assuming that the hypotenuse is the side with largest length so as per Pythagoras Theorem,

\((Hypotenuse)^{2} = (Perpendicular)^{2} + (Base)^{2}\)

5mm, 1.2mm, 0.9mm

Let \(\triangle PQR\) be the right angles triangle with \(\angle Q = 90^{\circ}\).

So, by Pythagoras theorem
\( 1.2^{2} + 0.9^{2} = 2.25\)\(PR^{2} = PQ^{2} + QR^{2}\)

\(? 1.5^{2} = 2.25 \)

R.H.S = L.H.S

Therefore, 1.5mm, 1.2mm, 0.9mmare

The lengths of the side of a right angled triangle.

3mm, 2mm, 1mm

Let \(\triangle PQR\) be the right angles triangle with \(\angle Q = 90^{\circ}\).

So, by Pythagoras theorem

\(PR^{2} = PQ^{2} + QR^{2}\) \(\ therefore 2^{2} + 1^{2} = 5\) \(? 3^{2} = 9 \) \(R.H.S \neq L.H.S\)

Therefore, 3mm, 2mm, 1mm are not

the lengths of the side of a right angled triangle.

1mm, 4mm, 0.9mm

Let \(\triangle PQR\) be the right angles triangle with \(\angle Q = 90^{\circ}\).

So, by Pythagoras theorem

\(PR^{2} = PQ^{2} + QR^{2}\) \(\ therefore 4^{2} + 0.9 ^{2} = 16.81\) \(? 4.1^{2} = 16.81 \)

R.H.S = L.H.S

Therefore, 4.1mm, 4mm, 0.9mm are

the lengths of the side of a right angled triangle.

Question 5:

A pole is broken at the height of 500cm from the footpath and its peak point is in contact with the footpath at a distance of 1200cm from the base of the pole. Find the actual length when it was in normal condition.

Answer: Let \(\triangle PQR\)be the right angles triangle with \(\angle Q = 90^{\circ}\) representing the pole broken at point P and touches the footpath at point R and Q is the base of the pole.

Here, PQ = 500cm, QR = 1200cm ,

Let PR = x cm

In \(\triangle PQR\) by applying Pythagoras theorem, we get

\(PR^{2} = PQ^{2} + QR^{2}\) \(\ therefore x^{2} =500^{2} + 1200^{2}\) \(?x^{2} = 250000 + 1440000 \) \(?x^{2} = 1690000\) \(? x = \sqrt{49} = 1300cm\)

Thus, PR = 1300cm

Now the actual length of the pole = PQ + PR

= 500 + 1300

= 1800cm

Thus, the actual length of the pole is 1800cm.

Question 6:

In \(\triangle XYZ\)

Find out the correct one from equations given below, \(\angle Y\; = 35^{\circ} and\;\angle Z=55^{\circ}\)

\(XZ^{2} = XY^{2} + YZ^{2}\)
\(XY^{2} = YZ^{2} + XZ^{2}\)
\(YZ^{2} = XY^{2} + XZ^{2}\)

Answer: In \(\triangle XYZ\)

\(\angle XYZ + \angle YZX + \angle YXZ = 180^{\circ}\)

\(35^{\circ} + 55^{\circ} +\angle YXZ = 180^{\circ}\)
\(\angle YXZ = 180^{\circ} – (35^{\circ} + 55^{\circ})\)
\(\angle YXZ = 180^{\circ} – 90^{\circ}\)
\(\angle YXZ = 90^{\circ}\)

So, \(\triangle XYZ\) is a right angled triangle with \(\angle X = 90^{\circ}\).

Thus, \((Hypotenuse)^{2} = (Perpendicular)^{2} + (Base)^{2}\)

\(YZ^{2} = XY^{2} + XZ^{2}\)

Thus, (c) is the correct option.

Question 7:

For a rectangle it is given that, width of a rectangle is 15mm and diagonal is 17 mm, then find the perimeter of the rectangle.

Answer: Let WXYZ be a rectangle.

Let the length of a rectangle is y mm.

In \(\triangle XYZ\) by applying

Pythagoras theorem, we get

\(XZ^{2} = YZ^{2} + XY^{2}\) \(? 17^{2} =15^{2} + y^{2}\) \(?y^{2} = 289 – 225 \) \(?y^{2} = 64\) \(? y = \sqrt{64} = 8mm\)

Thus, XY = 8mm

Thus, the length of a rectangle is 8mm.

Now, perimeter of rectangle = 2 (XY + YZ) = 2(8 + 15)

= 46mm

Thus, the perimeter of the rectangle is 46mm.

Question 8:

For a rhombus the length of diagonals are given 80mm and 18mm.Then determine the perimeter of the rhombus.

Answer: Let PQRS be the rhombus.

Let the length of side of rhombus is x mm.

Also PR = 18mm, SQ = 80mm

As the diagonals bisect each other at right angle in case of rhombus,

So, OP = PR/2 = 18/2 = 9mm

And SO = SQ/2 = 80/2 = 40mm

In \(\triangle POS\) by applying

Pythagoras theorem, we get

\(PS^{2} = OP^{2} + SO^{2}\) \(?x^{2} =9^{2} + 40^{2}\) \(?x^{2} = 81 + 1600 \) \(?x^{2} = 1681\) \(? x = \sqrt{1681} = 41mm\)

Thus, PS = 41mm

Now, the perimeter of the rhombus is = 4\(\times\)Length of side of rhombus = 4\(\times\)PS

= 4\(\times\)41 = 164mm

Thus, the perimeter of the rhombus is 164mm.