## Class 8: Triangles – Exercise 30C

Q2. State giving reasons, whether the following pairs of triangles are congruent or Not.

Answer

Congruent by ASA

Answer

Answer

ASA Theorem applied, Triangles are Congruent.

Answer

Applying Pythagoras theorem,

Hence Triangles are congruent by SAS

Q3. In the adjoining figure, P in the mid point of AB and . Prove that:

Answer:

, (given)

(given)

(opposite angles)

Hence

Q4. In the adjoining figure, . Prove that

Answer

Given

Therefore, by ASA

Q5. In the adjoining figure,

Answer

In

and DA = CB

Therefore, by ASA,

Q6. In the adjoining figure, ABC is a triangle in which .

Answer

Take

is common and

Therefore, by ASA,

Q7.

Answer

By SSS,

Therefore

Q8. In , it in given that in bisector of , meeting BC at D. Prove: i)

Answer

(angles opposite equal sides of a triangle)

(angle bisector)

Therefore by ASA:

Hence

Q9. In the adjoining figure, in such that . Prove that:

Answer

Given

Therefore,

AB = AC (given)

AO is Common

Because

Therefore, SAS applies, Hence

Since

Q10. In the adjoining figure,

Answer

In

(alternate angles)

(alternate angles)

Hence,

Given is common.

Therefore,

Applying ASA,

Q11. In the adjoining figure, ABCD as a square and CEB is an isosceles triangle in which EC = EB show:

Answer

Given (Sides of a square)

(Sides of an isosceles triangle)

Hence SAS applies,

Therefore,

Q12. Find the vales of x and y in each of the following cases:

Answers

i)

Given

(alternate angles)

Therefore, AAS applies Hence,

Therefore

ii)

In

Therefore by SSS,

Hence,

iii)

In

And PQ in common

Therefore, by SSS,

Hence

Therefore

Since,

Now Calculate,

iv)

In

Hence, ASA applies, Therefore,

Hence

And

Q13. In adjoining figure, the sides BA and CA of have been produced to D and E such that BA = AD and CA = AE Prove ED ∥ BC

Answer

In

(Opposite Angles)

Hence

Hence

Q14. Equilateral triangle ABC and ACE have been drawn on the sides AB and AC respectively of , as shown in the adjoining figure, prove: i)

Answer

In

Hence Therefore,

Q15. In a regular pentagon ABCDE, prove that is isosceles.

Answer

Since ABCD in a regular pentagon, all sides are equal and all internal angles are .

In

Therefore

Hence Therefore, isosceles triangle

Q16. In the adjoining figure, in a square, and are points on the side respectively such that,

Answer

Given ABCD in a square

(given)

Similarly

Now Consider

(Square)

Q17. In the adjoining figure,

Answer:

Since

And Similarly,

Given

Therefore, using AAS,

Q18. In the adjoining figure ABCD is a square, , and R is mid point of EF. Prove:

Answer

In ,

(Median would bisect the angle)

Common

In

(Side of a square)

(Proved above)

Because

Applying SAS, Proves that

Therefore, . Hence Proven.

Q19. In the adjoining figure, in the Mid point of PQ Prove:

Answer

In

(given)

(given)

Since

Q20. In adjoining figure,

Answer

Consider

is common

(given)

Hence,

Therefore,

Now Consider

is common

Therefore,

## Class 8: Triangles – Exercise 30D

Q.1 , right angled at A, find BC when:

i)

ii)

iii)

iv)

Q.2. In , right angled at Q, find PQ

i)

ii)

iii)

iv)

Q.3. The length of the side of some triangles is given below which one of them is right angled? In case of a right angled triangle, find which angle measures

i)

ii)

Not a right angled triangle

iii)

iv)

Not a right angled triangle

Q.4. Find the unknown side of each of the following figures;

Q.5. If the adjoining figure, it is given that:

.Find the length of

Answer:

Q.6. If the length of the diagonal of a rectangle is 37 cm. if the length of the shorter side is 12 cm, Find: i) The length of its longer side ii) Perimeter of rectangle iii) Area of rectangle

Answer:

Perimeter

Area

Q.7. The Base of an 150 sales triangle in 28 cm long and . Let find i) Length of AD ii) Area of ABC

Answer:

Area of ABC

Q.8. A diagonal of a rhombus in 16 cm long and each of its aides measures 10 cm. Find the length of the other diagonal.

Answer:

Diagonal of a rhombus bisect each other and also interest each other at right angle.

Shorter diagonal = 12 cm.

Q.9. The supporting wire to the top of a vertical pole in 13 cm long and it in fastened to the ground at a state 5 m among from the foot of the pole. How high is the pole?

Answer:

Let AB be the pole.

Q.10. A ladder 26 m long rust against a vertical wall with its foot 10m against from the wall how high up the wall will the ladder reach.

Answer:Let AB be the height of the ladder.

Q.11. A 15 meter against a vertical wall 21 reach a window at a height of 12m from the ground. How for in the foot of the ladder?

Answer

Let the distance of the foot of the ladder from the wall=d

Q.12. The height of thus to were are 34m and 10m respectively. If the distance between there fact in 32m, find the distance between their tops.

Answer:

Let l be the distance between the tops

Q.13. In the adjoining figure, ABC in a triangle in which . If D in the mid point of BC. Prove that

Answer:

Q.14. In the adjoining figure, it in given of that

Answer:-

Q.15. In the following figure, it in being given that:.

Find the Perimeter of:

Answer:

Perimeter of

Perimeter of

Perimeter of

Q.16. In the adjoining figure,

Answer:

Subtract ii) from i) ,

Q.17. In the Adjoining figure,

Answer:

Q.18. In a , if are mid point of , respectively and

Answer:

Adding (1) & (2)

Q.19. In quadrilateral

Answer:

Subtracting ii) from i)

## Class 8: Perimeter and Area of Planes (Lecture Notes)

TRIANGLES

__Perimeter of a Triangle__

If are the lengths of the sides of any triangle. Then:

- Perimeter of a triangle units
- Area of the triangle

where .This is also known as

__Area of a Triangle__

Refer to the adjoining figure. If b is the base and h is the height, then

Note: If you could take any side as the base, then the corresponding height is the

would be the length of the perpendicular to this side from the opposite vertex.

Let the triangle be ABC, with B = 90. Please refer to the adjoining figure.

__Area of an Equilateral triangle__

In an equilateral triangle, all three sides are equal.

Let us say the side unit.

- Height of an equilateral triangle = units
- Area of an equilateral triangle with side units

__RECTANGLE & SQUARE__

__Perimeter and Area of Rectangle__

If the sides of a rectangle are units and units (refer to adjoining figure), then

- Perimeter of rectangle units
- Area of rectangle sq. units
- Diagonal of a rectangle units

__Perimeter and Area of a Square__

If the sides of a square is units, then

- Perimeter units
- Area units
- Diagonal of a square units

__PARALLELOGRAM, RHOMBUS AND TRAPEZIUM__

__Area of a Parallelogram__

Let ABCD be a Parallelogram with base b and height h units. Let AC be the diagonal. Refer to the adjoining figure.

sq. units.

__Area of a Rhombus__

Please refer to the adjoin diagram. Let are the diagonals of the Rhombus. We know that the diagonals intersect at right angles and bisect each other.

sq. units

__Area of a Trapezium__

Please refer to the adjoining figure.

sq. units

__CIRCLE__

__Circumference and Area of a Circle__

Let the circle be of radius r

- Circumference of the circle
- Area of the circle

__Area of a Ring (shaded area)__

Refer to the adjoining figure. The radius of the larger circle is R and that of the smaller circle is r. Area of the ring is the shaded area.

## Class 8: Volume and Surface Area of Solids – Exercise 37

Q.1. Find the volume, the total surface area and the lateral surface area of the cuboid having:

__Answer:__

a)

Volume of a cuboid

Total surface Area of a cuboid

Lateral surface Area of a cuboid

b)

Volume of a cuboid

Total surface Area of a cuboid

Lateral surface Area of a cuboid

Q.2. Find the capacity of a rectangular tub whose length , breadth and depth . Also find the area of the iron sheet required to make the tub.

__Answer:__

Volume of the tub

Total surface Area of a cuboid

Q.3. A wall of length , width and height is to be constructed by using bricks of dimensions . How many bricks would be needed.

__Answer:__

Volume of the wall

Volume of the brick

Number of bricks needed

Q.4. How many planks each measuring can be stored in a place deep?

__Answer:__

Volume of the place

Volume of the plank

Number of planks stored

Q.5. A classroom is height. If each student is given of the floor area, how many students can be accommodated in the room? How many cubic meters of air would each student get?

__Answer:__

Area of the floor of the classroom

Area given to each student

Number of students that can be accommodated in the room

Cubic meters of air would each student get

Q.6. Find the length of the longest pole that can be placed in a room broad and high.

__Answer:__

Diagonal of a cuboid is the longest pole that can be placed in the room

Q.7. The volume of the cuboid is . If its length and breadth be respectively, find its height.

Volume of a cuboid

Q.8. The volume of the cuboid is . Its length is and its breadth and height Are in the ratio of . Find the breadth and height of the cube.

__Answer:__

Volume of a cuboid

Q.9. The surface area of the cuboid is . Its length and breadth are respectively. Find its height.

__Answer:__

Surface Area of a cuboid

Q.10. The length, breadth and height of the room are respectively. Find: i) the area of the four walls of the room ii)the area of the floor of the room.

__Answer:__

- i) Area of four walls would be

- ii) The area of the floor of the room

Q.11. A room high has one door and two windows each . Find the: i) area of four walls, excluding the doors and the windows. ii) cost of painting the wall from inside at a rate of . iii) the cost of painting the ceiling at .

__Answer:__

- i) Area of walls excluding the doors are windows

ii) Cost of painting the wall

iii) Costof painting the ceiling

Q.12. An assembly hall is height. It has five doors, each measuring and four windows each. Find the

- i) cost of wall paper at a rate of
- ii) cost of carpeting the floor at the rate of .

__Answer:__

Wall dimensions:

Door dimensions

Window dimensions

Area of walls excluding the doors are windows

- i) Cost of painting the wall
- ii) cost of carpeting the floor

The length, breadth and height of the cuboid are in the ratio of . If the surface area of the cuboid is , find its dimensions. Also find the volume of the cuboid.

__Answer:__

Wall dimensions:

Surface Area of a cuboid

Volume

Q.13. If the area of the three adjacent faces of a cuboidal box are respectively, then find the volume of the box.

Let the dimensions:

Multiplying the above three expressions we get

Q.14. A river deep and wide is flowing at a rate of . How many cubic meters of water runs into the sea per minute?

__Answer:__

Rate of flow

Volume of water flowing

Q.15. A closed wooded box high, is made up of wood 2.5 cm thick. Find i) the capacity of the box, ii) weight of the box if of wood weighs grams.

__Answer:__

External Volume of the Box

Internal Length

Internal Breadth

Internal Height

Internal Volume

Volume of Wood

The external dimensions of a wooden box, open at the top are . It is made up of wood thick. Calculate i) the capacity of the box ii) the volume of the wood.

__Answer:__

External Volume of the Box

Internal Length

Internal Breadth

Internal Height

Internal Volume

Volume of Wood

Q.16. The internal dimension of the closed box, made up of iron thick, are by . Find the volume of the iron in the box.

__Answer:__

Internal Volume of the Box

External Length

External Breadth

External Height

External Volume

Volume of Iron

Q.17. Find the volume, the total surface area and the lateral surface area and the diagonal of each cube whose edges measures: i) ii) iii)

__Answer:__

i)

Volume of a cube

Total surface Area of a cube

Lateral surface Area of a cube

Diagonal of a cube

ii)

Volume of a cube

Total surface Area of a cube

Lateral surface Area of a cube

Diagonal of a cube

iii)

Volume of a cube

Total surface Area of a cube

Lateral surface Area of a cube

Diagonal of a cube

Q.18. The surface area of the cube is . Find its volume.

__Answer:__

Surface Area of a cube

Therefore Volume of cube

Q.19. The volume of the cube is . Find its surface area.

__Answer:__

Volume of a cube

Surface area

Q.20. The volume of a cube is . Find its surface area.

Volume of a cube

Surface area

Q.21. A solid piece of metal in the form of cuboid of dimensions is melted down and re-casted into a cube. Find the length of each edge of the cube.

__Answer:__

Volume of a cuboid

Let the dimension of cube

Volume of Cube

Q.22. Three cubes of metal with edges are melted to form a single cube. Find the lateral surface area of the new cube formed.

__Answer:__

Let the dimension of the large cube

Volume of Large Cube

Therefore the dimension of the large cube

Lateral surface Area of a cube

## Class 8: Circle – Exercise 33

* *

Q.1. Fill in the blanks

- A line segment joining any point on the circle to its center is called a
*radius*of the circle. - All the radii of a circle are
.__equal__ - A line segment having its end points on a circle is called
of a circle.__chord__ - A chord that passes through the center of the circle is called a
of the circle.__diameter__ - Diameter of a circle is
its radius.__twice__ - A diameter is the
chord of the circle.__largest__ - The interior of a circle together with the circle is called the
circle.__area of the__ - A chord of a circle divides the whole circular region into two parts, each called a
.__segment__ - Half of a circle is called a
.__semicircle__ - A segment of a circle containing the center is called the
of the circle.*major segment* - The mid point of the diameter of a circle is the
of the circle.__center__ - The perimeter of the circle is called its
.__circumference__

* *

Q.2. State which of the following statements are true or false:

- Diameter of a circle is a part of a semi-circle of a circle :
*True* - Two semi-circles of a circle together make the whole circle:
*True* - Two semi-circular regions of a circle together make the whole circular region:
*True* - An infinite number of chords may be drawn in a circle:
*True* - A line can meet a circle at the most at two points:
*True* - An infinite number of diameters can be drawn in a circle:
*True* - A circle has an infinite number of radii:
*True* - A circle consists of an infinite number of points:
*True* - Center of a circle lies on a circle:
*True*

Q.3. From an external point P, 29 cm away from the center of a circle, a tangent PT of length 21 cm is drawn. Find the radius of the circle.

Answer:

Radius

Q.4. Two tangents are drawn from an exterior point p to a circle with center O. Prove that:

Answer:

In

PO is common, OM=ON (radius of the circle) and

(Tangents to a circle from one point circle are equal.)

Hence

Q.5. In the given figure, is inscribed in a circle with center O. If , find angle

Answer:

(angle in a semicircle is a right angle).

Q.6. In the given figure, O is the centre of a circle. is inscribed in this circle. If .

Answer:

(angle in a semicircle is a right angle).

Q.7. In the given figure, O is the centre of a circle. If , find . Also, if , find .

Answer:

(angle in a semicircle is a right angle).

Similarly

(angle in a semicircle is a right angle).

Q.8. In the given figure, is inscribed in a circle with center O. If , find the value of .

Answer:

(angle in a semicircle is a right angle).

Q.9. In the adjoining figure, PRT is a tangent to the circle with center O. QR is a diameter of the circle. If , then find the value of x.

Answer:

(tangent is perpendicular to the line drawn from the center to the point of contact)

Q.10. In the adjoining figure, PRT is a tangent to the circle with centre O. OR is the radius of the circle at the point of contact. P, O are joined and produced to the point Q on the circle. If , then find the values of .

Answer:

(tangent is perpendicular to the line drawn from the center to the point of contact)

Also

Q.11. In the adjoining figure, PT is a tangent to the circle with center O, QT is a diameter of the circle. If , then find the value of .

Answer:

Given

Q.12. In the adjoining figure, PX and PY are tangents drawn from an exterior point P to a circle with centre O and radius 8 cm. If PX = 15 cm, OP = a cm, PY = b cm, then find the value of .

Answer:

Q.13. In the adjoining figure, AB is the diameter of the circle with centre O. If , then find the value of .

Answer:

Q.14. In the adjoining figure, PQ is a diameter of a circle with center O. is an isosceles triangle with RP=RQ. PQ is produced to a point S such that RQ = QS. If , then find the values of .

Answer:

In

And we know

Given

## Class 8: Area Propositions-Exercise 32

Q.1. In the given figure, is right angled at B in which BC = 15 cm and CA = 17 cm. Find the area of acute angled , it being given that

Answer:

(since is a right angled triangle)

Area of

Q.2. In the adjoining figure, area

Answer:

Area of

This is because are between the same parallels and have the same base.

Also, since are on the same base and between the same parallels, so the

Area of

Q.3. In the adjoining figure, area

Answer:

Also, since are on the same base and between the same parallels, so the

Area of

Area of

This is because are between the same parallels and have the same base.

Q.4. In trapezium , it is being given that and diagonals intersect at O. Prove that:

Answer:

Since are on the same base and between the same parallels, the area of the two ‘s will be equal.

Also Since are on the same base and between the same parallels,the area of the ‘s will be equal.

Now, subtract the area of from both sides we get

Q.5. In the adjoining figure, is a parallelogram, P is a point on . Prove that : are equal in area

Answer:

Since are on the same base and between the same parallels, so the

Area of

Similarly, are on the same base and between the same parallels, so the

Area of

Area

Hence Proved.

Q.6. In the adjoining figure, is a quadrilateral. A line through D, parallel to AC, meets BC produced in P. Prove that .

Answer:

Similary,

Hence

But

Hence Proved that

Q.7. is any quadrilateral. Line segments passing through the vertices are drawn parallel to the diagonals of this quadrilateral so as to obtain a parallelogram as shown in the adjoining figure. Prove that:

Answer:

Since

Hence Proven.

## Class 8: Triangles – Exercise 30 B

Q.1. Find the value of in each of the following figures:

Answer:

i)

Similarly,

ii)

We know

Q.2. State giving reasons, whether it is possible to construct a triangle or not with sides of lengths:

Answer (Note: The sum of any two sides of a triangle is always greater than the third side)

i) Not Possible to construct the triangle because (3+4 not greater than 7

ii) Possible to construct the triangle as sum of any two sides is greater then the third side.

iii) Possible to construct the triangle as sum of any two sides is greater then the third side.

iv) Not possible to construct a triangle as sum of

Q.3. In . Find . Name i) the largest side of , ii) the smallest side of iii) write the sides of in ascending order of their lengths.

Answer:

. Therefore

i) the largest side of

ii) the smallest side of

iii)

Q.4. In . Name i) the smallest side of , ii) the largest side of iii) write the sides of in ascending order of their lengths.

Answer:

. Therefore

i) the largest side of

ii) the smallest side of

iii)

Q.5. In . Name the smallest side and the equal sides.

Answer

Smallest side

Equal Sides

Q.6 In . Name the largest side and the equal sides.

Answer:

Largest side , Equal Sides

Q.7. In . Name the smallest side and the largest sides.

Answer:

Largest side , Smallest Side

Q.8. In the adjoining figure, . Find . Also show that

Answer:

Therefore

Since

Since

Since

Q.9. In the adjoining figure, . If , find the values of .

Answer:

Since

(opposite angles)

Therefore

Which implies that

Becasue

Q.10. In the adjoining figure, . If , find the value of .

Answer:

Since

Therefore

Q.11. In the adjoining figure, and . Show that .

Answer:

Since

Since

Since

Since

Since

Q.12. If is a point on side , prove that:

Answer:

In

In

In , because it is a valid triangle,

Add the above two relations

Q.13. In the adjoing figure, is a quadrilateral. Prove that:

Answer:

In

In

Adding the above two expresions we get

. Add these two expressions we get