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Construction in mathematics refers to the construction of geometric figures using various geometric tools such as compass, protractor, ruler, etc. Construction is used in various forms in mathematics be it trigonometry, algebra or geometry. Through constructions students learn to draw a point by internally dividing a line segment through a given ratio, construct a tangent to a circle, construct triangles, , and construct a pair of tangents that are inclined to each other at an angle and so on.
MCQs for Constructions
Ques. For a line segment to be divided AB in the ratio 3:4, firstly a ray AX has been drawn so that ∠BAX come out as acute angle and then at equivalent distances points have been marked on the ray AX so now state what is the minimum number of these points ?
(1) 5
(2) 7
(3) 9
(4) 11
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Ans: (2) 7
Explanation: As we already know that for a line segment to be divided in the ratio m: n, firstly draw a ray AX which creates an acute angle BAX, then we have to mark m + n points at equivalent distances from each other.
So, m = 3, n = 4
Therefore, the minimum number for these points equals to i.e, m + n = 3 + 4 = 7
Ques. To divide a line segment AB of length 7.6 cm in the ratio 5 : 8, a ray AX is drawn first such that ∠BAX forms an acute angle and then points A1, A2, A3, ….are located at equal distances on the ray AX and the point B is joined to:
(1) A5
(2) A6
(3) A10
(4) A13
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Ans: (4) A13
Explanation: The minimum points placed in the ray AX is 5 + 8 = 13. Therefore, point B will have to join point A13.
Ques.To construct a pair of tangents to a circle at an angle of 60° to each other, it is needed to draw tangents at endpoints of those two radii of the circle, the angle between them should be:
(1) 100°
(2) 90°
(3) 180°
(4) 120°
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Ans: (4) 120°
Explanation: Since the figure generated by the intersecting point of two tangents and the two ends of those two radii and the axis of the circle is a quadrilateral, the angle between two radii ought to be 120°. As a result, the sum of the opposing angles should be 180 °.
Ques. To divide a line segment PQ in the ratio m : n, where m and n are two positive integers, draw a ray PX so that ∠PQX is an acute angle and then mark points on ray PX at equal distances such that the minimum number of these points is:
(1) m + n
(2) m – n
(3) m + n – 1
(4) Greater of m and n
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Ans: (1) m + n
Explanation : As we already know that for a line segment PQ to be divided in the ratio m: n, firstly draw a ray PX which creates an acute angle ∠ PQX, then we have to mark points on ray PX at equivalent distances and therefore the minimum number of these points is m + n.
Ques. To draw a pair of tangents to a circle which are inclined to each other at an angle of 45°, it is required to draw tangents at the endpoints of those two radii of the circle, the angle between which is:
(1) 135°
(2) 155°
(3) 160°
(4) 120°
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Ans. (1) 135°
Explanation : For drawing tangents on a circle that are inclined to one another at a given angle of 45°, tangents must be drawn at the endpoints of the circle’s two radii, the angle between which is 135°.
Ques. A pair of tangents can be constructed from a point P to a circle of radius 3.5 cm situated at a distance of ___________ from the centre.
(1) 3.5 cm
(2) 2.5 cm
(3) 5 cm
(4) 2 cm
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Ans: (3) 5 cm
Explanation: As the tangent pair can only be drawn with an external point, its distance from the center must have to be greater than that of the radius. And also only 5cm is exceeding the radius of 3.5cm. As a result, the tangents can be drawn from a point 5 cm from the center.
Ques. If the scale factor is 3/5, then the new triangle constructed is _____ the given triangle.
(1) smaller than
(2) greater than
(3) overlaps
(4) congruent to
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Ans: (1) smaller than
Explanation : As the scale factor is given 3/5, the newly constructed triangle have to be smaller than the provided triangle because the numerator is less than the denominator here.
Ques. By geometrical construction, which one of the following ratios is not possible to divide a line segment?
(1) 1 : 10
(2) √9 : √4
(3) 10 : 1
(4) 4 + √3 : 4 – √3
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Ans: (4) 4 + √3 : 4 – √3
Explanation : As the ratio 4 + √3 : 4 – √3 can not be simplified in the form of integers as other provided ratios, it is not possible to divide a line segment.
Ques. By geometrical construction, is it possible to divide a line segment in the ratio 1/√3 : √3?
(1) Yes
(2) No
(3) Cannot be determined
(4) None of these
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Ans: (1) Yes
Explanation:
Provided ratio is 1/√3 : √3
Multiplying by √3, we get,
(1/√3) × √3 : √3 × √3
1 : 3
So, the simplified ratio consists of only integers.
Therefore, with the help of geometrical construction it can be possible for given line segment to be divided.
Ques. In constructions, the scale factor is used to construct ______ triangles.
(1) right
(2) equilateral
(3) similar
(4) congruent
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Ans: (3) similar
Explanation: The scale factor is used to build similar triangles in constructions. Essentially, the scale factor specifies the ratio of the triangle sides to be formed to the corresponding triangle sides.
Ques. In the division of a line segment AB, any ray AX making angle with AB is _______.
(1) an acute angle
(2) a right angle
(3) an obtuse angle
(4) reflex angle
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Ans: (1) an acute angle.
Explanation : In division of a line segment AB, any ray AX making angle with AB is an acute angle.
Ques. A point P is at a distance of 8 cm from the centre of a circle of radius 5 cm. How many tangents can be drawn from point P to the circle?
(1) 0
(2) 1
(3) 2
(4) Infinite
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Ans: (3) 2
Explanation:
As Given,
Distance of a point from the center of the circle is more than that of Radius of the circle
Therefore, the point lies outside the circle.
So due to this, we are able to draw 2 tangents to the circle from the point P.
Ques. A line segment drawn perpendicular from the vertex of a triangle to the opposite side is known as
(1) altitude
(2) median
(3) bisector of side
(4) radius of incircle of the triangle
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Ans: (1) altitude
Explanation : Altitude is a line segment drawn perpendicular from the vertex of a triangle to the opposite end.
Quest. If the line segment is divided in the ratio 3 : 7, then how many parts does it contain while constructing the point of division?
(1) 3
(2) 7
(3) 4
(4) 10
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Ans: (4) 10
Explanation: The line segment is divided in three parts on one side and seven parts on the other side of the point of division, which implies it has three parts on one side and seven parts on the other. As a result, there will be a total of 10 parts as (3 + 7) = 10.
Ques. A pair of tangents can be constructed from a point P to a circle of radius 3.5 cm situated at a distance of ___________ from the center.
(1) 3.5
(2) 2.5
(3) 5
(4) 2
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Ans: (3) 5
Explanation: Only an external point can be used to draw the pair of tangents, therefore its distance from the center must be greater than the radius. So because the radius of 3.5cm is only 5cm larger than the radius of 5cm. As a result, the tangents can be drawn at a point 5cm away from the center.
Ques: To construct a triangle similar to a given ΔPQR with its sides, 9/5 of the corresponding sides of ΔPQR draw a ray QX such that ∠QRX is an acute angle and X is on the opposite side of P with respect to QR. The minimum number of points to be located at equal distances on ray QX is:
(1) 5
(2) 9
(3) 10
(4) 14
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Ans: (2) 9
Explanation: The lowest number of points to be positioned at an identical distance to draw a triangle similar to a specified triangle with sides m/n of the same sides of a given triangle is equivalent to m or n, whichever one is greater.
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