Free Constructions 01 Practice Test - 9th Grade
Question 1
In which of the following situations an angle bisector cannot be constructed to bisect the angle formed between two given lines?
SOLUTION
Solution : B
Since two parallel lines do not intersect each other, an angle cannot be formed between them. Thus in this case, angle bisection is not possible.
Question 2
In which of the following quadrilaterals, a diagonal is not an angle bisector?
SOLUTION
Solution : B
In the case of a rectangle, the diagonal is not an angle bisector.
Consider a rectangle ABCD.
Here, ∠ ADB = ∠ DBC (Alternate angles)
But, ∠ DBC is not equal to ∠ BDC (Angles opposite to unequal sides of a triangle are unequal)
∴ ∠ ADB is not equal to ∠ BDC
So, diagonal DB does not bisect ∠ D.
Question 3
Each point on a/an _________ is such that it forms an isosceles triangle with the end points of the given line segment.
SOLUTION
Solution : A
Consider the above figure.
Here, XY is the perpendicular bisector of a line AB.
Let P be any random point on XY.
In △ PMA ≅ △ PMB
AM = BM (Perpendicular bisector divides a line segment into two equal halves)
∠ PMA = ∠ PMB = 90∘
Also PM is common side
So △ PMA ≅ △ PMB (SAS Rule)
∴ PA = PB (CPCT)
Hence, △ PAB is an isosceles triangle.
So, the given statement is true.
Question 4
For which of the following can a perpendicular bisector be drawn?
Line
Ray
Line segment
Both Line and Ray
SOLUTION
Solution : C
A perpendicular bisector can be drawn only if a figure has end points. Only a line segment has a definite length and hence it can be bisected by a perpendicular bisector.
Question 5
Which among the following angles cannot be constructed just by using a ruler and a compass?
30 degrees
60 degrees
70 degrees
90 degrees
SOLUTION
Solution : C
All the standard angles and the angles which can be obtained by bisecting standard angles can be constructed just by using a ruler and compass. Here except 70∘ all other angles can be constructed using ruler and compass.
Question 6
For constructing a right angle without a protractor, which of the following method/s is/are used?
Angle Bisector
60∘ angle construction
Construction of triangle
SOLUTION
Solution : A and B
To construct a right angle triangle, the following steps should be followed:
1. Draw a line OB of given length.
2. With O as centre, make an arc of any radius intersecting OB at X.
3. With X as a centre, draw another arc keeping the radius same intersecting the previous arc at D.
4. ∠DOB will be 60∘.
5. With D as a centre and keeping the radius same, draw another arc intersecting the first arc at C.
6. ∠COB will be 120∘ .
7. Bisect COD drawing two equal arcs from each points intersecting at E.
8. Join EO and extend till A. ∠AOB will be 90∘.
So, it can be seen that both 60∘ and angle bisector construction is used in the process.
Question 7
Two sides of a triangle have lengths of 5 units and 6 units respectively. Which of the following is a possible length for the third side?
11
12
13
10
SOLUTION
Solution : D
In a triangle, the sum of the lengths of any 2 sides of a triangle must be greater than the third side. The third side can measure anything less than 11 units. Hence, the third side can be 10 units.
Question 8
For constructing a triangle with a given base, a base angle and difference between the other two sides, a
SOLUTION
Solution :The procedure for Triangle Construction 2 is:
1. Draw the base BC of ∆ABC as given and construct ∠XBC of the required measure at B as shown.
2. From the ray, BX cut an arc equal to AB – AC at point P and join it to C as shown
3. Draw the perpendicular bisector of PC and let it intersect BX at point A as shown:
4. Join AC, ∆ABC is the required triangle.
Hence, it is clear that a perpendicular bisector is required in the process.
Question 9
A triangle ABC has base angle 45∘. It's perimeter is [2+√2]. What type of triangle is it?
Isosceles triangle
Right triangle
Can't be determined
Scalene triangle
SOLUTION
Solution : C
The information given above is insufficient to construct a triangle. So, the triangle cannot be constructed and its characteristics can’t be determined.
Question 10
The construction of ΔABC, given that BC = 12 cm, is possible when the difference of AB and AC equals
12.5 cm
8 cm
SOLUTION
Solution : B and C
The difference between two sides of a triangle should be smaller than the third side. So only 8 cm and 10 cm can be the possible difference.