NCERT Class 10 Chapter 15 Constructions CBSE Board Sample Problems Long Answer (For CBSE, ICSE, IAS, NET, NRA 2022)

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Question

Draw two concentric circles of radii 3 cm and 5 cm. construct a tangent to smaller circle from a point on the larger circle. Also measure its length

Solution

Now after measuring, PA and PB comes out to be 4 cm.

Steps of construction of tangents:

Take point O. Draw 2 concentric circles of radii 3 cm and 5 cm respectively.

Locate point P on the circumference of larger circle.

Join OP and bisect it. Let M be mid-point of OP.

Taking M as center and MP as radius, draw an arc intersecting smaller circle at A and B.

Join PA and PB. Thus, PA, PB are required tangents

Question

Construct a in which AB = 6 cm, and . Construct another

similar to with base .

Solution

Steps of construction:

Draw and make an angle of at point B and at point A.

Make any acute angle BAX at point A.

Cut four arcs on line AX such that

Cut four arcs on line AX such that Join B to .

Draw line from parallel to cutting AB extended to B′.

Draw line from cuts AC at C′.

Steps of construction:

1. Draw a circle of radius 4 cm with center O.

2. Take point A on circle. Join OA.

3- Draw line AP perpendicular to radius OA.

4. Draw at O.

5. Join A and B at P, to get 2 tangents. Here

Question

Draw an isosceles in which and altitude. Then construct another triangle whose sides are of the corresponding sides of

Solution

Steps of construction:

Draw BC = 5.5 cm.

Construct AP the perpendicular bisector of BC meeting BC at L.

Along LP cut off LA = 3 cm.

Join BA and CA. Then MBC so obtained is the required .

Draw an acute angle CBY and cut 4 equal lengths as and join .

Now draw a line through parallel to intersecting BC at C′.

Draw a line through C′ and parallel to AC intersecting AB at A′. BA′C′ is the required triangle.

Question

Draw a triangle with sides 5 cm, 6 cm, and 7 cm. Then draw another triangle whose sides are

of the corresponding sides of first triangle

Solution

Draw a line segment AB of length 7 cm.

Then using A as center and distance 5 cm draw an arc C.

Also draw an arc using B as center and with distance 6 cm. which intersect earlier drawn arc at C. Join AC and BC.

Draw an acute angle BAZ and cut AZ as and join .

Through draw a line parallel BA-5 intersecting AB at B′.

Through B′ draw a line parallel to BC intersecting AC at C′. AAB′C′ is the required triangle.

Question

Construct a triangle ABC with BC = 7 cm, and . Construct another triangle whose sides are times the corresponding sides of

Solution

Steps of construction:

Draw a line segment BC = 7 cm.

Draw at point B. Thus

Take an arc of 6 cm, with B as center mark an arc on BX to get point A.

Join AC.

is constructed triangle.

Draw an acute angle CBY below BC.

Take points at BY such that

Join P-4-C with line.

Draw a line parallel to through the point which intersects BC at C′.

Join with line

Question

Construct an isosceles triangle whose base is 6 cm and altitude 4 cm. Then construct another triangle whose sides are times the corresponding sides of the isosceles triangle.

Solution

Steps of construction:

Draw a line segment

Draw perpendicular bisector of BC which intersects BC at point D.

Take an arc of 4 cm, with D as center mark on arc Bisector as point A.

Join AB and AC. is constructed isosceles .

Draw an acute angle CBY below BC.

Take points at BY such that

Join P-4-C with dotted line.

Draw a line parallel to through the point which intersects BC at C′.

Join with dotted line.

Draw a line parallel to AC through the point C′ which intersects AB at A′.

is the required triangle whose sides are times the corresponding sides of

Question

Draw a line segment AB of length 7 cm. Taking A as center, draw a circle of radius 3 cm and taking B as center, draw another circle of radius 2 cm. Construct tangent to each circle from the center of the other circle?

Solution

Required tangents are

BP and BQ

AR and AS.

Steps of construction:

Draw AB = 7 cm. Taking A and B as centers, draw two circles of 3 cm and 2 cm radius.

Bisect line AB. Let mid-point of AB be C.

Taking C as center, draw circle of AC radius which will intersect circles at P, Q, R, S.

Join BR BQ, AR, AS

Question

Construct a in which AB = 6 cm, and . Construct another

Similar to with base ?

Solution

Steps of construction:

Draw AB = 6 cm and make an angle of at point B and at point A.

Make any acute angle BAX at point A.

Cut four arcs on line AX such that

Join B to A-3.

Draw line from parallel to cutting AB extended to B′.

Draw line from cuts AC at C′.

Question

Construct a right triangle ABC with , and . Draw BD, the perpendicular from B on AC. Draw the circle through B, C and D and construct the tangents from A to this circle.

Solution

Thus, AP and AB are the required tangents

Steps of construction:

Draw .

Take an arc of 6 cm, with B as centre, mark an arc on point A. Join AB.

Draw . Bisect line BC at E as mid-point of BC.

Taking E as centre and EC as its radius, draw circle which will intersect AC at D. Join BD.

Mark point P on circle. Join A to P.

Question

Construct a triangle ABC in which AB = 5 cm, BC = 6 cm and . Now construct another triangle whose sides are times the corresponding sides of .

Solution

Steps of construction:

Draw BC = 6 cm.

Take any radius (less than half of BC) and center B, draw an arc intersecting BC at

P. With same radius and center P, draw another arc intersecting previous arc at Q.

Join BQ, extend it to D.

Take radius = 5 cm and center B, we draw arc intersecting BD at A. Join AC, get .

Draw line BX, as is any acute angle. Draw 7 equal radius arcs on line BX intersecting at as

Join to C. Draw line from as parallel to intersecting BC at C′.

Draw line from C′ as parallel to AC intersecting AB at A′.

Question

Draw a circle of radius 4.8 cm. Take a point P on it. Without using the center of the circle, construct a tangent at the point P. Write the steps of construction.

Solution

Steps of Construction:

Step 1. Draw a circle of radius 4.8 cm.

Step 2. Mark a point P on it.

Step 3. Draw any chord PQ.

Step 4. Take a point R on the major arc QP.

Step 5. Join PR and RQ.

Step 6. Draw

Step 7. Produce TP to T′, as shown in the figure.

TPT is the required tangent.

Question

Draw a circle of radius 4 cm. Draw a tangent to the circle, making an angle of with a line passing through the center.

Solution

Steps Of construction:

Step 1. Draw a circle with center O and radius 4 cm.

Step 2. Draw radius OA and produce it to B.

Step 3. Make .

Step 4. Draw . meeting OB at Q.

Step 5. Then, PQ is the desired tangent, such that .

Question

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

Solution

Steps of Construction

Step 1. Mark a point O on the paper.

Step 2. With O as center and radii 4 cm and 6 cm, draw two concentric circles.

Step 3. Mark a point P on the outer circle.

Step 4. Join OP.

Step 5. Draw the perpendicular bisector XY of OR cutting OP at Q.

Step 6. Draw a circle with Q as center and radius OQ (or PQ) , to intersect the inner circle in points T and T.

Step 7. Join PT and PT′.

Here, PT and PT are the required tangents.

(Approx.)

Verification by actual calculation

Join OT to form a right (Radius is perpendicular to the tangent at the point of contact)

In right ,

(Pythagoras Theorem)

Question

Draw a circle of radius 3 cm. Draw a tangent to the circle making an angle of with a line passing through the center.

Solution

Steps Of construction:

Step 1. Draw a circle with center O and radius 3 cm.

Step 2. Draw radius OA and produce it to B.

Step 3. Make .

Step 4. Draw . meeting OB at Q.

Step 5. Then, is the desired tangent, such that

Question

Draw a circle of radius 4.2 cm. Draw a pair of tangents to this circle inclined to each other at an angle of Draw a circle of radius 4.2 cm. Draw a pair of tangents to this circle inclined to each other at an angle of .

Solution

Steps of Construction:

Step 1. Draw a circle with center O and radius = 4.2 cm.

Step 2. Draw any diameter AOB of this circle.

Step 3. Construct , such that the radius OC meets the circle at C.

Step 4. Draw and .

AM and CN intersect at P.

Thus, PA and PC are the required tangents to the given circle inclined at an angle of .

Question

Draw a line segment AB of length 8 cm. Taking A as center, draw a circle of radius 4 cm and taking B as center, draw another circle of radius 3 cm. Construct tangents to each circle from the center of the other circle.

Solution

Steps of Construction

Step 1. Draw a line segment AB = 8 cm.

Step 2. With A as center and radius 4 cm. draw a circle.

Step 3. With B as center and radius 3 cm. draw another circle.

Step 4 Draw the perpendicular bisector XV of AB. cutting AB at C.

Step 5. With C as center, and radius AC (or BC) , draw a circle intersecting the center, with center A at P and P′; and the circle with center B at O and O′.

Step 6. Join BP and BP′. Also, join AQ and AO′.

Here, AQ and AQ ′ are the tangents from A to the circle with center B. Also. BP aid BP are the tangents from Bio the circle with center A.

Question

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

Solution

Step 1. Draw a circle with the help of a bangle.

Step 2. Mark a point P outside the circle.

Step 3. Through P, draw a secant PAB to intersect the circle at A and B.

Step 4. Produce AP to C such that PA = PC.

Step 5. Draw a semicircle with CB as diameter.

Step 6. Draw , intersecting the semicircle at D.

Step 7. With P as center and PD as radius, draw arcs to intersect the circle at T and T.

Step 8. Join PT and PT′.

Here, PT and PT′ are the required pair of tangents.

Question

Draw a circle with center O and radius 4 cm. Draw any diameter AB of this circle. Construct tangents to the circle at each of the two end points of the diameter AB.

Solution

Steps of Construction

Step 1. Draw a circle with center O and radius 4 cm.

Step 2. Draw any diameter AOB of the circle.

Step 3. At A, draw . Produce XA to Y.

Step 4. At B, draw Produce X′B to Y′.

Here, XAY and X′BY′ are the tangents to the circle at the end points of the diameter AB.

Question

Draw a circle of radius 3.5 cm. Take two points A and B on one of its extended diameter, each at a distance of 5 cm from its center. Draw tangents to the circle from each of these points A and B.

Solution

Steps of Construction

Step 1. Draw a circle with center O and radius 3.5 cm.

Step 2. Extends its diameter on both sides and mark t points A and B on it such that OA = OB = 5 cm.

Step 3. Draw the perpendicular bisectors of OA and OB. Let C and D be the mid-points of OA and OB, respectively.

Step 4. Draw a circle with C as center and radius OC (or AC) , to intersect the circle with center O at the points P and Q.

Step 5. Draw another circle with D as center and radius OD (or BD) , to intersect the circle with center O at the points R and S.

Step 6. Join AP and AQ. Also join BR and BS.

Here, AP and AQ are the tangents to the circle from A. Also, BR and BS are the tangents to the circle from B.

Question

Draw a circle of radius 3 cm. Take two points P and Q on one of its diameters extended on both sides, each at a distance of 7 cm on opposite sides of its center.

Draw tangents to the circle from these two points P and Q.

Solution

Step 1. Draw a circle with O as center and radius 3 cm.

Step 2. Mark a point P and Q on one of its diameters extended on both sides outside the circle such that OP = OQ = 7 cm.

Step 3. Join OP and OQ. Draw the perpendicular bisector XY of OP and X′Y′ of OQ, cutting OP at L and OQ at M.

Step 4. Draw a circle with L as center and radius PL (or OL) . to intersect the given circle at the points A and B. Draw another circle with M as center and radius MQ (or OM) , to intersect the given circle at the points C and D.

Step 5. Join PA and PB. Join QC and QD

Here, PA, PB and QC, QD are the required tangents.

Question

Draw a line segment of length 7.6 cm and divide it in the ratio 5: 8. Measure the two parts.

Solution

Step 1. Draw a line segment AB = 7.6 cm

Step 2. Draw a ray AX, making an acute angle .

Step 3. Along AX, mark points and such that

Step 4. Join .

Step 5. From , draw parallel to (draw an angle equal to ) , Meeting AB in R

Here, P is the point on AB which divides it in the ratio 5: 8.

Length of AP = 2.9 cm (Approx)

Length of BP 4.7 cm (Approx.)

Question

Draw a line segment of length 8 cm and divide it internally in the ratio 4: 5.

Solution

Step 1. Draw a line segment AB = 8 cm

Step 2. Draw a ray AX, making an acute angle .

Step 3. Along AX, mark points and such that

Step 4. Join .

Step 5. From draw parallel to (draw an angle equal to ) , meeting AB in D.

Here, D is the point on AB which divides it in the ratio 4: 5.

Question

Draw two tangents to a circle of radius 3.5 cm from a point P at a distance of 6.2 cm from its center.

Solution

Step 1. Draw a circle with O as center and radius 3.5 cm.

Step 2. Mark a point P outside the circle such that OP = 6.2 cm.

Step 3. Join OP. Draw the perpendicular bisector XY of OP, cutting OP at Q.

Step 4. Draw a circle with Q as center and radius PQ (or OQ) , to intersect the given circle at the points T and T′.

Step 5. Join PT and PT′.

Here, PT and PT′ are the required tangents.

Question

Construct a in which and Construct another simitar to with base .

Solution

Step 1. Draw a line segment

Step 2. At A, draw .

Step 3. At B, draw Suppose AX and BY intersect at C.

Thus, is the required triangle.

Step 4. Produce AB to B′ such that AB′ = 8 cm.

Step 5. From B′, draw meeting AX at C′.

Here, is the required triangle similar to .

Question

Construct a, in which PQ = 6 cm, QR = 7 cm and PR = 8 cm. Then, construct another triangle whose sides are times the corresponding sides of .

Solution

Step 1. Draw a line segment QR = 7 cm.

Step 2. With Q as center and radius 6 cm, draw an arc.

Step 3. With R as center and radius 8 cm, draw an arc cutting the previous arc at P.

Step 4. Join PQ and PR. Thus, is the required triangle.

Step 5. Below QR, draw an acute angle .

Step 6. Along QX, mark five points and such that

Step 7. Join .

Step 8. From , draw meeting QR at R′.

Step 9. From R′, draw meeting PQ in P′.

Here, is the required triangle, each of whose sides are times the corresponding sides of .

Question

Draw a line segment AB of length 7 cm. Using ruler and compasses, find a point P on AB such that

Solution

Step 1. Draw a line segment AB = 7 cm.

Step 2. Draw a ray AX, making an acute angle .

Step 3. Along AX, mark 5 points (greater of 3 and 5) and such that

Step 4. Join .

Step 5. From , draw parallel to (draw an angle equal to ) , meeting AB in P.

Here, P is the point on AB such that