Class 10 Maths - Constructions

Draw a line segment of length 10 cm. Divide it into four equal parts. Measure each of these parts.

Steps of construction:
Step I - draw a line segment AB = 10 cm with the help of a ruler.

Step II - With vertex A, by using compass take more than half of AB and draw arcs, one each side of sides of AB

Step III - With vertex B, by using a compass and with the same radius cut the previous arcs on both sides of AB at R and S respectively. Join RS.

Mark the point O on AB where RS intersects it.

Step IV - With vertex A, by using compass take more than half of AO and draw arcs, one each side of sides of AO

Step V - With vertex O, by using a compass and with same radius cut the previous arcs on both sides of AO at P and Q respectively. Join PQ.

Mark the point M on AB where PQ intersects it.

Step VI - With vertex O, by using compass take more than half of OB and draw arcs, one each side of sides of OB

Step VII - With vertex B, by using a compass and with the same radius cut the previous arcs on both sides of OB at T and U respectively. Join TU.

Mark the point N on AB where TU intersects it.

Each part measures

$+2.5\ cm$

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Draw a circle of radius $3\ cm$ . Take two points $P$ and $Q$ on one of its extended on both sides of its centre, each at a distance of $7\ cm$ on opposites of its centre. Draw tangents to the circle from each of these two points $P$ and $Q$

Step 1: Draw a circle with centre

$O$ and radius

$3\ cm$ using a compass.

Step 2: Draw a secant passing through the centre. Mark points

$P$ and

$Q$ on opposite sides of the centre at a distance of

$7\ cm$ from

$O$ .

Step 3: Place the compass on

$P$ and draw two arcs on opposite sides of

$OP$ . Now place the compass on

$O$ and draw two arcs intersecting the arcs drawn from point

$P$ .

Step 4: Join the intersection points of the arcs to obtain the perpendicular bisector of

$OP$ . Mark the mid point of

$OP$ as

$M_1$

Step 5: From

$M_1$ draw a circle with radius

$=M_1P= M_1O$

Step 6: Mark the intersection points of the circle drawn from

$M_1$ with the circle drawn from

$O$ as

$A$ and

$B$ .

Step 7: Join

$P-A$ and

$P-B$

Step 8: Repeat steps

$3$ to

$7$ for point

$Q$ and obtain tangents

$QC$ and

$QD$

$PA,PB,QC$ and

$QD$ are the required tangents.

Construct a tangent to a circle of radius $4cm$ from a point on the concentric circle of radius $6cm$ and measure its length. Also verify the measurement by actual calculation.

Construction the two conuntric circle with to

$r_1 = 4 cm$ and

$r_2 =6cm$ with centre be

$C$

locate a point

$P$ on outer circle such that

$CP= 6cm$ and lets assume

$M$ be the mid point of CP such that

$CM=MP=3cm$

Now draw a circle with centre

$M$ and radius

$= 2 cm$ and lets say its intersect the inner circle at point

$A$ and

$B$ .

Now join the points

$A,P$ and

$B,P$ so we get our tangents

$AP$ and

$BP$

Since

$\triangle CAP$ is a right angled triangle with

$\angle A = 90^o$

therefore using pythogorus therom

$(CA)^2+(AP)^2=(CP)^2$

$(PA)^2=36-9$

$AP=3\sqrt{3} cm$

Draw a circle with the help of a bangle. Take a point $P$ outside the circle. Construct the pair of tangents from this point $P$ to the circle.

Step 1: Draw a circle by tracing the surface of a bangle.

Step 2: Take a point

$P$ outside the circle. Draw secant

$P-R-Q$

Step 3: Place the compass on

$P$ and mark two arcs on opposite side of

$PQ$ . Now, place the compass on

$Q$ and draw two arcs intersecting the arcs drawn from

$P$ .

Step 4: Join the intersection points of the arcs to obtain the perpendicular bisector of

$PQ$ . Mark the mid point of

$PQ$ as

$M$ .

Step 5: Place the compass on

$M$ and taking

$PM$ as radius draw a semicircle passing through

$P$ and

$Q$ .

Step 6: From point

$R$ , draw a perpendicular to

$PQ$ and let it intersect the semicircle drawn in step

$5$ at

$C$ .

Step 7: Place the compass at

$P$ and with

$PC$ as radius, mark two arcs on the original circle and mark them as

$A$ and

$B$ .

Step 8: Join

$P-A$ and

$P-B$

$PA$ and

$PB$ are the required tangents.

Draw a triangle $ABC$ with side $BC = 6$ cm, $\angle C = 30^o$ and $\angle A = 105^o$ . Then construct another triangle whose sides are $\dfrac{2}{3}$ times the corresponding sides of $\Delta ABC$ .

Given:

$BC = 6$ cm,

$\angle C= 30^o$ and

$\angle A = 105^o$
We know that in

$\Delta ABC, \angle A + \angle B + \angle C = 180^o$
Therefore,

$\angle B = 180^o - (\angle A + \angle C)$

$= 180^o − (105^o + 30^o)\Rightarrow 45^o$
Steps of construction:
1. Draw a line

$BC = 6$ cm.
2. Draw a ray CN making an angle of

$30^o$ at

$C.$
3. Draw a ray BM making an angle of

$45^o$ at

$B.$
4. Locate the point of intersection of rays

$CN$ and

$BM$ and name it as

$A.$
5.

$ABC$ is the triangle whose similar triangle is to be drawn.
6. Draw any ray

$BX$ making an acute angle with

$BC$ on the side opposite to the vertex

$A.$
7. Locate 3 (Greater of

$2$ and

$3$ in

$\dfrac{2}{3}$ ) points

$B_1, B_2$ and

$B_3$ on

$BX$ so that

$BB_1 = B_1B_2 = B_2B_3$ .
8. Join

$B_3C$ and draw a line through

$B_2$ (Smaller of

$2$ and

$3$ in

$\dfrac{2}{3}$ ) parallel to

$B_3C$ to intersect

$BC$ at

$C'.$
9. Draw a line through

$C'$ parallel to the line

$CA$ to intersect

$BA$ at

$A'.$
10.

$A'BC'$ is the required similar triangle whose sides are

$\dfrac{2}{3}$ times the corresponding sides of

$\Delta ABC$ .

Constructions - Class 10 Maths - Extra Questions

Draw a line segment of length 10 cm. Divide it into four equal parts. Measure each of these parts.

Draw a circle of radius $3\ cm$ . Take two points $P$ and $Q$ on one of its extended on both sides of its centre, each at a distance of $7\ cm$ on opposites of its centre. Draw tangents to the circle from each of these two points $P$ and $Q$

Construct a tangent to a circle of radius $4cm$ from a point on the concentric circle of radius $6cm$ and measure its length. Also verify the measurement by actual calculation.

Draw a circle with the help of a bangle. Take a point $P$ outside the circle. Construct the pair of tangents from this point $P$ to the circle.

Draw a triangle $ABC$ with side $BC = 6$ cm, $\angle C = 30^o$ and $\angle A = 105^o$ . Then construct another triangle whose sides are $\dfrac{2}{3}$ times the corresponding sides of $\Delta ABC$ .

Class 10 Maths Extra Questions

Constructions - Class 10 Maths - Extra Questions

Draw a line segment of length 10 cm. Divide it into four equal parts. Measure each of these parts.

Draw a circle of radius 3 cm3\ cm. Take two points PP and QQ on one of its extended on both sides of its centre, each at a distance of 7 cm7\ cm on opposites of its centre. Draw tangents to the circle from each of these two points PP and QQ

Construct a tangent to a circle of radius 4cm4cm from a point on the concentric circle of radius 6cm6cm and measure its length. Also verify the measurement by actual calculation.

Draw a circle with the help of a bangle. Take a point PP outside the circle. Construct the pair of tangents from this point PP to the circle.

Draw a triangle ABCABC with side BC=6BC = 6 cm, ∠C=30o\angle C = 30^o and∠A=105o\angle A = 105^o. Then construct another triangle whose sides are 23\dfrac{2}{3} times the corresponding sides of ΔABC\Delta ABC.

Class 10 Maths Extra Questions

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Draw a circle of radius $3\ cm$ . Take two points $P$ and $Q$ on one of its extended on both sides of its centre, each at a distance of $7\ cm$ on opposites of its centre. Draw tangents to the circle from each of these two points $P$ and $Q$

Construct a tangent to a circle of radius $4cm$ from a point on the concentric circle of radius $6cm$ and measure its length. Also verify the measurement by actual calculation.

Draw a circle with the help of a bangle. Take a point $P$ outside the circle. Construct the pair of tangents from this point $P$ to the circle.

Draw a triangle $ABC$ with side $BC = 6$ cm, $\angle C = 30^o$ and $\angle A = 105^o$ . Then construct another triangle whose sides are $\dfrac{2}{3}$ times the corresponding sides of $\Delta ABC$ .