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Practice problems on Basic proportionality theorem
Section A: Multiple Choice Questions
1. Basic proportionality theorem was introduced by a famous
greek mathematician named
(a) Aryabhata
(b) Thales
(c) Brahmagupta
(d) Bhaskara
2. Two similar triangles MNO and DEF are given. Which of the
following formulae can be used to find out the ratio of the
areas of the given triangles.
(a) The ratio of the areas of the triangles is equal to the square
of the corresponding medians
(b) The ratio of the areas of the triangles is equal to the square
of the corresponding altitudes
(c) All of these can be used to evaluate the ratio of the areas of
the triangles
(d) None of the above
3. From the below given figure, which of the following options
is correct for the image to satisfy basic proportionality theorem.
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(a) ABC is an equilateral triangle
(b) D,E are midpoints of AB and AC respectively
(c) AEBD
(d) DEBC
4. Which among the following similarity criterion cannot be
used for triangles
(a) SAS
(b) SSS
(c) RHS
(d) ASA
5. Which similarity can be used if 2 angles and 1 side of a
triangle are equal
(a) SAS
A
B
C
D
E
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(b) ASA
(c) SA
(d) AA
6. In the below figure, DEBC. Find the value of x.
(a)
4
5
(b)
3
5
(c)
4
7
(d)
2
5
7. In a triangle ABC, points P, Q being on AB and AC
respectively. Which among the following conditions is sufficient
to prove that the triangle does not satisfy basic proportionality
theorem.
A
B
C
D
E
x-5
2x+2
x+1
x-2
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(a) Length of AB and AC is equal
(b) PQ is not parallel to BC
(c) P and Q are midpoints on the sides
(d) Triangle is a isosceles triangle
8. In a triangle ABC, a line PQ is drawn parallel to BC, points P, Q
being on AB and AC respectively. If AB=4AP, then what is the
ratio of the area of triangle APQ to the area of triangle ABC?
(a) 4:1
(b) 1:4
(c) 16:1
(d) 1:16
9. The areas of two similar triangles are 16
2
and 25
2
,
respectively. The ratio of their corresponding heights is
(a) 4:5
(b) 5:4
(c) 3:4
(d) 5:6
10. In a triangle ABC, if PQBC and P intersect on AB and Q
intersect on AC. If AB=14cm, AQ=7 and QC=8. Find the value of
AP.
(a) 7.9
(b) 8
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(c) 8.2
(d) 8.3
Section B
11. Explain Basic proportionality theorem.
12. Find the length EB from the given figure if FEBC.
13. In ΔABC, if D and E are mid point of BC and AD respectively
such that ar(AEC)=4
2
then ar(BEC)=
14. Prove that If the areas of two similar triangles are equal
then the triangles are congruent.
A
B
C
E
F
6
5
8
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15. In a triangle XYZ, a line MN is drawn parallel to YZ, points M,
N being on XY and XZ respectively. If XY=2XM, then what is the
ratio of the area of triangle XMN to the area of triangle XYZ?
Section A(Answers)
1. Option (b) Thales is the correct answer
Basic proportionality theorem was introduced by a famous
greek mathematician named Thales.
2. Option (c) is the correct answer
The ratio of the areas of the triangles is equal to the square of
the corresponding medians.
The ratio of the areas of the triangles is equal to the square of
the corresponding altitudes.
Any of these can be used to find the ratio of given triangles.
3. Option (d) DEBC is the correct answer
DEBC is the condition for the triangle to satisfy basic
proportionality theorem.
4. Option (c) RHS is the correct answer
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RHS is not a criteria which is used to find the similarity of
triangles.
5. Option (b) ASA is the correct answer
If 2 angles and 1 side are similar then the triangle satisfies
angles, side, angle property.
6. Option (a)
is the correct answer
From the above figure
=
.
5
+ 1
=
+ 2
2
(5)(2) = (+ 2)(+ 1)
2
7+ 10 =
2
+ 3+ 2
10= 8
=
4
5
.
7. Option (b) is the correct answer
If the side PQ is not parallel to BC then the triangle does not
satisfy basic proportionality theorem.
8. Option (d) 1:16 is the correct answer
If PQ is parallel to BC then we can say that the triangle satisfies
basic proportionality theorem.
=
2
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=
4
2
=
1
16
Ratio of the area of triangle APQ to the area of triangle
ABC=1:16
9. Option (a) 4:5 is the correct answer
Given that the area of 2 similar triangles is in the ratio 16:25
For similar triangles
1
2
=
2
16
25
=
1
2
2
16
25
=
1
2
Therefore the sides are in the ratio 4:5.
10. Option (c) 8.2 is the correct answer
According to basic proportionality theorem
We know that
=
Given AB=14, AQ=7, QC=5
AC=AQ+QC=12
Substituting the values
=
14
=
12
5
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=
5
14
12
= 5.8
We know that
AB=AP+PB
14=AP+5.8
AP=8.2
Section B(Answers)
11.
Basic proportionality theorem states that if a line is drawn
parallel to one side of a triangle which meets the other 2 sides
in distinct points, then the other 2 sides are divided in the same
ratio.
12. 6.67 is the correct answer
Given FEBC
According to BPT
=
5
=
6
8
=
8
5
6
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= 6.67.
13.
If the areas of two similar triangles are equal then the triangles
are congruent and they have equal length for the
corresponding sides.
14.
Let the 2 triangles be ABC and DEF
We know that for similar triangles ratio of area of the triangles
is equal to the square of the ratios of the corresponding sides
=
2
=
2
=
2
Given that area are equal
So 1 =
2
=
2
=
2
From the above equation, we can say that
AB=DE, BC=EF, AC=DF
Hence Proved.
15. 4:1 is the correct answer
Using BPT, we know that
=
2
XY and XM are corresponding sides in that triangle.
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=
2
2
=
2
1
2
Therefore the areas of triangles are in the ratio 4:1.
