CBSE Class 9 Maths Lab Manual – Areas of Two Triangles on the Same Base and between the Same Parallels

Objective
To show that the triangles on the same base and between the same parallel lines are equal in area experimentally.

Prerequisite Knowledge

1. Familiarity with triangles.
2. Formula for area of triangle = $$\frac{ 1 }{ 2 }$$ x base x height.
3. Shortest distance between the two parallel lines.

Materials Required
Glazed papers, a pair of scissors, pencil, gluestick, white sheet.

Procedure

1. Draw any triangle on a glazed paper and name it as ABC.
2. Cut the triangle and paste it on the white sheet as shown in fig.(i).
3. Cut another triangle EGH such that EH = BC [fig .(ii)]
4. Draw a line l at point A such that l is parallel to BC as shown in fig.(iii).
5. Draw any triangle KBC with the base BC and its vertex K lying on the line l as shown in fig (iii).
6. Paste ∆EGH on ∆ABC such that EH lies on BC and G does not lie on line l as shown in fig (iii).

Observation
ar(∆ABC) = $$\frac{ 1 }{ 2 }$$ x base x height
= $$\frac{ 1 }{ 2 }$$ x BC x (⊥distance between || lines l and BC)
= $$\frac{ 1 }{ 2 }$$ x BC x AM
ar(∆BKC) = $$\frac{ 1 }{ 2 }$$ x BC x (⊥distance between || lines l and BC)
= $$\frac{ 1 }{ 2 }$$ x BC x KL
= $$\frac{ 1 }{ 2 }$$ x BC x AM (⊥distance between || lines is always same), (KL = AM)
∴ ar(∆ABC) = ar(∆BKC)
ar(∆EGH) = $$\frac{ 1 }{ 2 }$$ x base x height
= $$\frac{ 1 }{ 2 }$$ x BC x (⊥distance from G to BC)
= $$\frac{ 1 }{ 2 }$$ x BC x GN
But, AM ≠ GN
∴ ar(∆ABC) ≠ ar(∆EGH)

Result
We have verified that two triangles on the same base and between the same parallel lines are equal in area.

Learning Outcome
We learnt that areas of two or more triangles are same if they lie on the same base and between the same parallel lines. Triangles having same base but different perpendicular heights are not same and their areas are also not equal.

Activity Time
Students can verify this theorem by graphical and counting method.
[Hint: Draw two triangles on the same base and between the same height and then count the squares covered by two triangles on the graph paper].

Viva Voce

Question 1.
Two triangles are on the same base and same height, area of one of the triangle is 4 cm2. What will be the area of second triangle ?
4 cm2

Question 2.
In a triangle, a median divides the triangle in two parts. What will be the area of two triangles, so formed ?
Area of two triangles will be same.

Question 3.
If two triangles are having same areas then the triangles will be congruent ?
Not necessary.

Question 4.
If two triangles are having same base 4 cm and their area is 12 cm2 and lying between the same parallel lines, what will be the height?
6 cm.

Question 5.
Parallelograms on the same base and having equal areas lie between the same parallels. Is the converse true ?
Yes, the converse is true.

Question 6.
If a triangle and a parallelogram are on the same base and between the same parallels, then what will be area of triangle ?
Area of triangle is equal to half the area of parallelogram.

Question 7.
In the above question, area of parallelogram is 12 cm2, what will be the area of the triangle ?
6 cm2

Question 8.
If a triangle and a parallelogram are on the same base and not lie between same parallel lines, then will they have same altitudes ?
No, they will not have same altitudes.

Question 9.
If two triangles are on same base and having same areas. What can you say about their heights.
Their heights will be same.

Multiple Choice Questions

Question 1.
Triangles ABC and DBC are on the same base BC and with vertices A and D on opposite sides of BC such that ar (∆ABC) = ar (∆DBC). What is the relation between their altitudes ?
(i) equal
(ii) not equal
(iii) altitude of ∆ABC = altitude of ∆BDC
(iv) none of these

Question 2.
D and E are points on sides AB and AC respectively of ∆ABC, such that ar(∆DBC) = ar(∆EBC). Are lines DE and BC parallel to each other ?
(i) yes
(ii) no
(iii) DE = BC
(iv) none of these

Question 3.
In ∆ABC, AB = 8 cm and altitude corresponding to AB is 5cm. In ∆DEF, EF = 10 cm. Find the altitude corresponding to side EF, if ar (∆ABC) = ar(∆DEF):
(i) 8 cm
(ii) 5 cm
(in) 4 cm
(iv) none of these

Question 4.
The area of rectangle ABCD is 50 sq. cm. If P be any point on AB, find area of ∆PCD :
(i) 25 sq.cm
(ii) 100 sq.cm
(iii) 50 sq. cm
(iv) none of these

Question 5.
Find area of triangle whose base is (x – 3) and height is (x + 4) and its area is x2/2:
(i) 72 cm2
(ii) 12 cm2
(iii) 144 cm2
(iv) none of these

Question 6.
In triangle ABC, O is the mid-point on its median AD. If area of ∆ABD is 12 cm2. Find area of ∆ABO:
(i) 12 cm2
(ii) 24 cm2
(iii) 6 cm2
(iv) none of these

Question 7.
In ∆ABC, D, E and F are mid-points of the sides BC, CA, AB respectively. If area of ∆ABC is 144 cm2. What will be the area of triangle DEF ?
(i) 36 cm2
(ii) 12 cm2
(iii) 72 cm2
(iv) none of these

Question 8.
In ∆PQR, M, N, P are mid-points of the sides PQ, QR, RP respectively. If area of ∆PQR is 162 cm2, then what will be the area of parallelogram PNQM ?
(i) 81 cm2
(ii) 41 cm2
(iii) 54 cm2
(iv) none of these

Question 9.
ABDE and BCDE are two parallelograms on the same base ED. AB = BC = DE. If ar(∆BDE) = 75 cm2. What will be area of ACDE ?

(i) 75 cm2
(ii) 225 cm2
(iii) 3 cm2
(iv) none of these

Question 10.
AD is a median of ∆ABC. If ar(∆ABD) = x cm2 and ar(∆ABC) = y cm2. Find the relation between x and y:
(i) x – 2y
(ii) y – 2x
(iii) y – x
(iv) none of these