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Maths 4 Mark Question

Geometrical Constructions · Maharashtra Board STD 6 (MSBSHSE - Semi English Medium). 15 questions.

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4 Mark Question

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15 questions · self-marked practice — reveal the answer and mark yourself.

Question 14 Marks
Construct a rectangle whose adjacent sides are $8cm$ and $3cm.$
Answer
Draw a line segment AB of length 8cm.
Construct $\angle\text{BAX}=90^{\circ}$ at point A and $\angle\text{ABY}=90^{\circ}$ at point B.
Using a compass and ruler, mark a point D on the ray AX such that AD = 3cm.
Similarly mark the point C on the ray Y such that BC = 3cm.
Draw the line segment CD.
ABCD is the required rectangle.
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Question 24 Marks
Construct the angle with the help of ruler and compasses only:
150°
Answer
Draw a line AB and take point O at the middle of AB.
With a convenient radius and centre at O, draw an arc, which cuts the line AB at P and Q.
With the same radius and centre at Q, draw an arc, which cuts the first arc at R.
With the same radius and centre at R, draw an arc, which cuts the first arc at S.
With the centres P and S and radius more than half of PS, draw two arcs, which cut each other at T.
Draw OT and extend it to C to form the ray OC.
$\angle\text{BOC}$ is required angle of 150°.
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Question 34 Marks
Draw a circle with centre at point O and radius 5cm. Draw its chord AB, draw the perpendicular bisector of line segment AB. Does it pass through the centre of the circle?
Answer
Draw a point O. With O as centre and radius equal to 5cm, draw a circle.
Take any two points A and B on the circumference of the circle and draw a line segment with A and B as its end points.
AB is the chord of the circle.
With A as centre and radius more than half of AB, draw arcs on both sides of AB.
With the same radius and B as a centre, draw arcs on both sides of AB, cutting the previous two arcs at E and F.
Draw a line passing through E and F.
Line EF passes through the centre of the circle O.
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Question 44 Marks
Construct the angle with the help of ruler and compasses only:
105°
Answer
Draw a ray OA and make an angle $\angle\text{AOB}=90^{\circ}$ and $\angle\text{AOC}=120^{\circ}$
Now bisect $\angle\text{BOC}$ and get the ray OD.
$\angle\text{AOD}$ is the required angle of 105º
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Question 54 Marks
Draw a line segment AB of length 5.8cm. Draw the perpendicular bisector of this line segment.
Answer
Draw a line segment AB of length 5.8cm using a ruler.
With A as centre and radius more than half of AB, draw arcs on both sides of AB.
With the same radius and B as centre, draw arcs on both sides of AB, intersecting the previous arcs at L and M.
Draw the line segment LM with L and M as end-points.
LM is the required perpendicular bisector of AB.
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Question 64 Marks
Construct an angle of 60° with the help of compasses and bisect it by paper folding.
Answer
Draw a ray OA. With convenient radius and centre O, draw an arc cutting the ray OA at P. With the same radius and centre at P, draw another arc cutting the previous arc at Q. Draw OQ and extend it to B. $\angle\text{AOB}$ is the required angle of 60°.
We cut the part of paper as sector OPQ. Now, fold the part of paper such that line segments OP and OQ get coincided. Angle made at point O is the required angle, which is half of angle $\angle\text{AOB}.$
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Question 74 Marks
Draw a line segment of length 8.6cm. Bisect it and measure the length of each part.
Answer
Draw a line segment AB of length 8.6cm.
With A as centre and radius more than half of AB, draw arcs on both sides of AB.
With the same radius and B as centre, draw arcs on the both sides of AB, cutting the previous two arcs at E and F.
Draw a line segment from E to F intersecting AB at C.
On measuring AC and BC, we get: AC = BC = 4.3cm.
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Question 84 Marks
Construct the angle with the help of ruler and compasses only:
135°
Answer
Draw the line AB and take the point O at the middle of AB.
With a convenient radius and centre at O, draw an arc, which cuts AB at P and Q, respectively.
Draw an angle of 90° on the ray OB as $\angle\text{BOC}=90^{\circ},$ where ray OC cuts the arc at R.
With Q and R as centres and radius more than half of QR, draw two arcs, which cuts each other at S.
Draw OS and extend it to form the ray OD.
$\angle\text{BOD}$ is required angle of 135°.
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Question 94 Marks
Construct two segments of lengths 4.3cm and 3.2cm. Construct a segment whose length is equal to the sum of the lengths of these segments.
Answer
Using compass and ruler, we construct two segments AB and CD of lengths 4.3cm and 3.2cm, respectively.
Draw a line L and mark a point P on it.
Take a compass and place its metal point at A and adjust it, such that the pencil point reaches point B.
Take the compass to line L, such that its metal point is on P.
Mark a small mark at Q on the line L corresponding to the pencil point of the compass.
Now, reset the compass, such that its metal and pencil points are on C and D, respectively.
Take the compass again to line L, such that its metal point is on Q and the pencil point makes a small mark at point R, which opposite to point P on line L
PR is the required segment, whose length is equal to the sum of the lengths of these segments.
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Question 104 Marks
Draw an obtuse angle. Bisect it. Measure each of the angles so obtained.
Answer
Obtuse angles are those angles which are greater than 90° but less than 180°.
Draw an obtuse angle $\angle\text{BAC}.$
With an appropriate radius and centre at A, draw an arc such that it intersects AB and AC at P and Q, respectively.

With centre P and radius more than half of PQ, draw an ARC.
With the same radius and centre at Q, draw another arc intersecting the previous arc at R.
Join A and R and extend it to X.
The ray AX is the required bisector of $\angle\text{BAC}.$
If we measure $\angle\text{BAR}$ and $\angle\text{CAR},$
we have $\angle\text{BAR}=\angle\text{CAR}=65^{\circ}$
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Question 114 Marks
Construct the angle with the help of ruler and compasses only:
30°
Answer
Draw a ray OA.
With a convenient radius and centre at O, draw an arc, which cuts OA at P.
With the same radius and centre at P, draw an arc cutting the previous arc at P.
Taking P and Q as centres and radius more than half of PQ, draw two arcs, which cuts each other at R.
Draw OR and extend it to B.
$\angle\text{AOB}$ is the required angle of 30°.
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Question 124 Marks
Construct the angle with the help of ruler and compasses only:
90°
Answer
Draw a ray OA.
With a convenient radius and centre at O, draw an arc cutting the ray OA at P.
With the same radius and centre at P, draw another arc, which cuts the first arc at Q.
With the same radius and centre at Q, draw another arc, which cuts the first arc at R.
With Q and R as centres and radius more than half of QR, which cuts each other at S.
Draw OS and extend it to B from the ray OB.
$\angle\text{AOB}$ is required angle of 90°.
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Question 134 Marks
Draw a linear pair of angles. Bisect each of the two angles. Verify that the two bisecting rays are perpendicular to each other.
Answer
Two angles, which are adjacent and supplementary, are called linear pair of angles.
Draw a line AB and mark a point O on it.
When we draw any angle $\angle\text{AOC},$ we also get another angle $\angle\text{BOC}.$
Bisect $\angle\text{AOC}$ by a compass and a ruler and get the ray OX.
Similarly, bisect $\angle\text{BOC}$ and get the ray OY.
Now,
$\angle\text{XOY}=\angle\text{XOC}+\angle\text{COY}$
$=\frac{1}2{}\angle\text{AOC}+12\angle\text{BOC}$
$=\frac{1}{2}(\angle\text{AOC}+\angle\text{BOC})$
$=\frac{1}{2}\times180^{\circ}=90^{\circ}$ (As $\angle\text{AOC}$ and $\angle\text{BOC}$ are supplementary angles)
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Question 144 Marks
Construct the angle with the help of ruler and compasses only:
45°
Answer
To construct an angle of 45°, construct an angle of 90° and bisect it.
Construct the angle $\angle\text{AOB}=90^{\circ},$ where rays OA and OB intersect the arc at points P and T as shown in figure.
With P and T as centres and radius more than half of PT, draw two arcs, which cut each other at X
Draw OX and extend it to C to form the ray OC.
$\angle\text{AOC}$ is the required angle of 45°.
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Question 154 Marks
Draw an angle and label it as $\angle\text{BAC}.$ Construct another angle, equal to $\angle\text{BAC}.$
Answer
Draw an angle $\angle\text{BAC}$ also draw a ray OP.
With a suitable radius and A as center, draw an arc intersecting AB and AC at X and Y, respectively.
With the same radius and O as center, draw an arc to intersect the arc OP at M.
Measure XY using the compass.
With M as centre and radius equal to XY, draw an arc to intersect the arc drawn from O at N.
Join 0 and N and extend it to Q.
$\angle\text{POQ}$ is the required angle.
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