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14
NCERT Solutions · Maths Ch 14
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36 questions · 5 exercises
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Ex 14.1
Ex 14.2
Ex 14.3
Ex 14.4
Ex 14.5
NCERT answers with full exam-style working, step by step. Tap a question to open its solution.
1
EXERCISE 14.1 · 6 QUESTIONS
Rectangles and squares
Q1
Identify the missing sidelengths. (i) Four rectangles are placed around a central gap. Their areas are 28 in², 21 in², 35 in² and 14 in². The known lengths are 3 in, 4 in, 7 in and 2 in, and the missing width of the 14 in² rectangle is marked '? in'. (ii) A rectangle of height 4 m is split into two rectangles of area 29 m² and 11 m² with their widths marked '?'; below the 29 m² rectangle is another rectangle of area 50 m² whose height is marked '?'.
(5 marks)
Q2
The figure shows a path (the shaded portion) laid around a rectangular park EFGH inside a rectangle ABCD. (i) What measurements do you need to find the area of the path? Once you identify the lengths to be measured, assign possible values of your choice to these measurements and find the area of the path. Give a formula for the area. (ii) If the width of the path along each side is given, can you find its area? If not, what other measurements do you need? Assign values of your choice to these measurements and find the area of the path. Give a formula for the area using these measurements. (iii) Does the area of the path change when the outer rectangle is moved while keeping the inner rectangular park EFGH inside it?
(5 marks)
Q3
The figure shows a plot with sides 14 m and 12 m, and with a crosspath. What other measurements do you need to find the area of the crosspath? Once you identify the lengths to be measured, assign some possible values of your choice and find the area of the path. Give a formula for the area based on the measurements you choose.
(3 marks)
Q4
Find the area of the spiral tube shown in the figure. The tube has the same width throughout (1 unit). The straight pieces along the spiral, from the inside out, measure 5, 5, 10, 10, 15, 15, 20, 20 and 20 units. What should be the length of the straight tube if it is to have the same area as a bent tube with arms 5 and 5?
(3 marks)
Q5
In this figure, if the sidelength of the square is doubled, what is the increase in the areas of the regions 1, 2 and 3? Give reasons. (The square has a diagonal; the midpoint of the diagonal is joined to a corner. This gives region 1 (a triangle), region 2 (a triangle) and region 3 (half of the square).)
(5 marks)
Q6
Divide a square into 4 parts by drawing two perpendicular lines inside the square as shown in the figure. Rearrange the pieces to get a larger square, with a hole inside.
(3 marks)
2
EXERCISE 14.2 · 8 QUESTIONS
Triangles
Q1
Find the areas of the following triangles: (i) ΔABC with base BC = 4 cm and altitude AE = 3 cm (ii) ΔDEF with base EF = 5 cm and altitude DN = 3.2 cm (iii) right-angled ΔNAT with AN = 4 cm and AT = 3 cm (right angle at A).
(3 marks)
Q2
Find the length of the altitude BY. (In ΔABC, AX = 4 units is perpendicular to the line of BC, BC = 6 units, AC = 8 units, and BY is perpendicular to AC.)
(3 marks)
Q3
Find the area of ΔSUB, given that it is isosceles, SE is perpendicular to UB, and the area of ΔSEB is 24 sq. units.
(2 marks)
Q4
[Śulba-Sūtras] Give a method to transform a rectangle into a triangle of equal area.
(3 marks)
Q5
[Śulba-Sūtras] Give a method to transform a triangle into a rectangle of equal area.
(3 marks)
Q6
ABCD, BCEF, and BFGH are identical squares. (i) If the area of the red region is 49 sq. units, then what is the area of the blue region? (ii) In another version of this figure, if the total area enclosed by the blue and red regions is 180 sq. units, then what is the area of each square?
(5 marks)
Q7
If M and N are the midpoints of XY and XZ, what fraction of the area of ΔXYZ is the area of ΔXMN? [Hint: Join NY]
(3 marks)
Q8
Gopal needs to carry water from the river to his water tank. He starts from his house. What is the shortest path he can take from his house to the river and then to the water tank? Roughly recreate the map in your notebook and trace the shortest path.
(3 marks)
3
EXERCISE 14.3 · 5 QUESTIONS
Area of any polygon
Q1
Find the area of the quadrilateral ABCD given that AC = 22 cm, BM = 3 cm, DN = 3 cm, BM is perpendicular to AC, and DN is perpendicular to AC.
(3 marks)
Q2
Find the area of the shaded region given that ABCD is a rectangle. (AE = 10 cm, EB = 8 cm, AF = 6 cm, FD = 4 cm; the shaded region is the quadrilateral FECD.)
(5 marks)
Q3
What measurements would you need to find the area of a regular hexagon?
(3 marks)
Q4
What fraction of the total area of the rectangle is the area of the blue region? (A rectangle with an inner point P; the top blue triangle has the top side as its base and P as its apex, and the bottom blue triangle has the bottom side as its base and P as its apex.)
(3 marks)
Q5
Give a method to obtain a quadrilateral whose area is half that of a given quadrilateral.
(3 marks)
4
EXERCISE 14.4 · 9 QUESTIONS
Parallelograms
Q1
Observe the parallelograms (a) to (g) drawn on the grid. (i) What can we say about the areas of all these parallelograms? (ii) What can we say about their perimeters? Which figure appears to have the maximum perimeter, and which has the minimum perimeter?
(3 marks)
Q2
Find the areas of the following parallelograms: (i) base 7 cm, height 4 cm (ii) base 5 cm, height 3 cm (iii) a side of 5 cm and the perpendicular to it of length 4.8 cm (iv) a side of 2 cm and the perpendicular to it of length 4.4 cm
(3 marks)
Q3
Find QN. (PQRS is a parallelogram with SR = 12 cm, PS = 7.6 cm, QM ⊥ SR with QM = 6 cm, and QN ⊥ PS.)
(3 marks)
Q4
Consider a rectangle and a parallelogram of the same sidelengths: 5 cm and 4 cm. Which has the greater area? [Hint: Imagine constructing them on the same base.]
(3 marks)
Q5
Give a method to obtain a rectangle whose area is twice that of a given triangle. What are the different methods that you can think of?
(3 marks)
Q6
[Śulba-Sūtras] Give a method to obtain a rectangle of the same area as a given triangle.
(3 marks)
Q7
[Śulba-Sūtras] An isosceles triangle can be converted into a rectangle by dissection in a simpler way. Can you find out how to do it? (ΔABC is isosceles with AB = AC; D is the midpoint of BC.) [Hint: Show that triangles ΔADB and ΔADC can be made into halves of a rectangle.]
(3 marks)
Q8
[Śulba-Sūtras] Give a method to convert a rectangle into an isosceles triangle by dissection.
(3 marks)
Q9
Which has greater area — an equilateral triangle or a square of the same sidelength as the triangle? Which has greater area — two identical equilateral triangles together or a square of the same sidelength as the triangle? Give reasons.
(3 marks)
5
EXERCISE 14.5 · 8 QUESTIONS
Rhombus and trapezium
Q1
Find the area of a rhombus whose diagonals are 20 cm and 15 cm.
(2 marks)
Q2
Give a method to convert a rectangle into a rhombus of equal area using dissection.
(3 marks)
Q3
Find the areas of the following figures: (i) a trapezium with parallel sides 10 ft and 7 ft and the perpendicular distance between them 16 ft (ii) a trapezium with parallel sides 24 m and 36 m and height 14 m (iii) a trapezium with parallel sides 14 in and 6 in and the perpendicular distance between them 10 in (iv) a trapezium with parallel sides 12 ft and 18 ft and height 8 ft
(5 marks)
Q4
[Śulba-Sūtras] Give a method to convert an isosceles trapezium to a rectangle using dissection.
(3 marks)
Q5
Here is one of the ways to convert trapezium ABCD into a rectangle EFGH of equal area. Given the trapezium ABCD, how do we find the vertices of the rectangle EFGH? [Hint: If ΔAHI ≅ ΔDGI and ΔBEJ ≅ ΔCFJ, then the trapezium and rectangle have equal areas.]
(3 marks)
Q6
Using the idea of converting a trapezium into a rectangle of equal area, and vice versa, construct a trapezium of area 144 cm².
(3 marks)
Q7
A regular hexagon is divided into a trapezium, an equilateral triangle, and a rhombus, as shown. Find the ratio of their areas.
(3 marks)
Q8
ZYXW is a trapezium with ZY ∥ WX. A is the midpoint of XY. Show that the area of the trapezium ZYXW is equal to the area of ΔZWB. (B is the point where ZA, extended, meets WX extended.)
(3 marks)