ABCD is a trapezium with parallel sides AB = a cm and DC = b cm (Fig. 9.6). E and F are the mid-points of the non-parallel sides. The ratio of ar (ABFE) and ar (EFCD) is
a. a : b
b. (3a + b) : (a + 3b)
c. (a + 3b) : (3a + b)
d. (2a + b) : (3a + b)
Solution:
It is given that
AB = a cm
DC = b cm
AB || DC
E and F are the mid-points of AD and BC
Consider h as the distance between AB, CD and EF
Now join BD which intersects EF at M
In ∆ ABD,
E is the midpoint of AD and EM || AB
M is the midpoint of BD
Using the midpoint theorem
EM = 1/2 AB … (1)
In ∆ CBD
MF = 1/2 CD … (2)
From equations (1) and (2)
EM + MF = 1/2 AB + 1/2 CD
EF = 1/2 (AB + CD)
EF = 1/2 (a + b)
Here
Area of trapezium ABFE = 1/2 [sum of parallel sides] × [distance between parallel sides]
Substituting the values
= 1/2 [a + 1/2 (a + h)] × h
= 1/4 [3a + b] h
Similarly
Area of trapezium EFCD = 1/2 [b + 1/2 (a + h)] × h = 1/4 [3b + a] h
Required ratio = Area of trapezium ABFE/ Area of trapezium EFCD
By substituting the values
= 1/4 [3a + b] h/ 1/4 [3b + a] h
= [3a + b]/ [3b + a]
= [3a + b]: [3b + a]
Therefore, the ratio of ar (ABFE) and ar (EFCD) is (3a + b) : (a + 3b).
✦ Try This: The figure obtained by joining the mid-points of the adjacent sides of a rectangle of sides 9 cm and 6 cm is :
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 9
NCERT Exemplar Class 9 Maths Exercise 9.1 Problem 10
ABCD is a trapezium with parallel sides AB = a cm and DC = b cm (Fig. 9.6). E and F are the mid-points of the non-parallel sides. The ratio of ar (ABFE) and ar (EFCD) is a. a : b, b. (3a + b) : (a + 3b), c. (a + 3b) : (3a + b), d. (2a + b) : (3a + b)
Summary:
ABCD is a trapezium with parallel sides AB = a cm and DC = b cm (Fig. 9.6). E and F are the mid-points of the non-parallel sides. The ratio of ar (ABFE) and ar (EFCD) is (3a + b) : (a + 3b)
☛ Related Questions:
- If P is any point on the median AD of a ∆ ABC, then ar (ABP) ≠ ar (ACP). Is the given statement true . . . .
- If in Fig. 9.7, PQRS and EFRS are two parallelograms, then ar (MFR) = 1/2 ar (PQRS). Is the given st . . . .
- ABCD is a parallelogram and X is the mid-point of AB. If ar (AXCD) = 24 cm² , then ar (ABC) = 24 cm² . . . .