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Constructions

Class 10th Mathematics Rajasthan Board Solution
Exercise 14.1
  1. Draw a line segment of length 6.7 cm and divide it internally in the ratio 2 :…
  2. Draw a line segment AB = 8.3 cm. Locate a point C on the line segment AB such…
  3. Draw a circle of radius 2.8 cm. Take a point P on it and draw a tangent to the…
  4. Construct tangents at the ends of the diameter of a circle of radius 3 cm. With…
  5. Draw a circle of radius 13.1 cm. Construct tangents to the circle at the ends…
  6. Draw a circle of radius 2.7 cm and draw a tangent to the circle.
  7. Take a point O and draw a circle of radius, 2.4 cm. Draw two radii OA and OB…
  8. Draw a pair of tangents to a circle of radius 13.2 cm which are inclined to…
  9. Draw a circle of radius 3 cm. Take a point P at a distance of 5 cm from the…
  10. The centers of two circles of radii 3 cm and 4 cm are 8 cm apart. How many…
  11. The radii of two circles are 1.7 cm and 2.8 cm, and their centers are 6 cm…
Exercise 14.2
  1. Write the truth value (T/F) of each of the following statements. If possible,…
  2. Draw incircle of an equilateral triangle of side 4.6 cm. Do its circumcentre…
  3. Draw incircle of a triangle ABC where AB = 4.6 cm, AC = 4.2 cm and ∠A = 90°.…
  4. Draw circumcircle of a triangle whose sides are 10.5 cm, 12.7 cm and 13 cm…
  5. Where should the circumcentre of a triangle be located if sides of the triangle…
  6. Construct ΔABC where AB = 6 cm, BC = 4 cm and ∠B = 120°. Also draw incircle of…
Exercise 14.1
Question 1.

Draw a line segment of length 6.7 cm and divide it internally in the ratio 2 : 3.
Answer:


Steps of Construction:

1. Draw a line segment AB of length 6.7cm.



2. Draw a ray AX making an acute angle with AB.



3. As we have to divide the line in ratio of 2:3, we will divide AX into five equal parts.


4. Taking A as center, cut an arc on AX. Mark the point as A1.


5. Now, with A1 as center and same radius cut an arc on AX. Continue till A5.



6. Join A5 to B.



7. Now draw a line from point A2 parallel to A5B.


8. The point where it cuts AB is P.



9. AP:PB=2:3
Question 2.

Draw a line segment AB = 8.3 cm. Locate a point C on the line segment AB such that Also justify it.
Answer:

1. Draw a line segment AB of length 8.3cm.



2. Draw a ray AX making an acute angle with AB.



3. As we have to divide the line such that AC = AB /3, we will divide AX in three equal parts.


4. Taking A as center, cut an arc on AX. Mark the point as A1.


5. Now, with A1 as center and same radius cut an arc on AX. Continue till A3.



6. Join A3 to B.



7. Now draw a line from point A1 parallel to A3B.


8. The point where it cuts AB is C.



9. AC:AB = 1:3
Question 3.

Draw a circle of radius 2.8 cm. Take a point P on it and draw a tangent to the circle at point P.
Answer:

Steps of Construction:


1. Draw a circle with center O and radius 2.8cm.


2. Take a point P on the circumference of this circle.



3. Join OP. Extend OP such that OP=PM.



4. Draw a perpendicular bisector of OM.


5. This perpendicular bisector is the tangent of the circle at P.
Question 4.

Construct tangents at the ends of the diameter of a circle of radius 3 cm. With the tangents intersect each other? Give reason for your answer.
Answer:

Steps of Construction:


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


2. Draw a diameter AB of this circle.



3. Extend OA such that OA=AP.


4. Extend OB such OB=BQ.



5. Draw a perpendicular bisector of OP.


6. This perpendicular bisector is the tangent at A.



7. Draw a perpendicular bisector of OQ.


8. This perpendicular bisector is the tangent at B.



No, these tangents will not intersect. They are parallel lines. Sum of internal angles is 180°. That’s why they are parallel.


∠CAO + ∠DBO = 90° + 90° = 180°
Question 5.

Draw a circle of radius 13.1 cm. Construct tangents to the circle at the ends of a chord of length 2.3 cm of the circle.
Answer:

1. Draw a circle with radius 13.1 cm.



2. Draw a chord AB = 2.3 cm. Then join B to center O.



3. Extend OA such that OA=AX. Similarly extend OB to OY.



4. Draw perpendicular bisector of OX.



5. Similarly, draw perpendicular bisector of OY.


Mark the intersection of perpendiculars as T.



Here, TB and TA are required tangents.
Question 6.

Draw a circle of radius 2.7 cm and draw a tangent to the circle.
Answer:

Steps of Construction:


1. Draw a circle with center O and radius 2.7cm.


2. Take a point P on the circumference of this circle.



3. Join OP. Extend OP such that OP=PM.



4. Draw a perpendicular bisector of OM.


5. This perpendicular bisector is the tangent of the circle at P.
Question 7.

Take a point O and draw a circle of radius, 2.4 cm. Draw two radii OA and OB inclined to each other at 60°. Construct tangents at points A and B which intersect at point T. Measure angle ATB.
Answer:

Steps of Construction:


1. Draw a circle with center O and radius 2.4cm.


2. Draw a radius OA of this circle.



3. Draw another radius OB which is 60° to OA.



4. Extend OA such that OA=AP.


5. Draw a perpendicular bisector of OP.



6. Extend OB such that OB=OQ.


7. Draw a perpendicular bisector of OQ.



8. The two perpendicular bisectors meet at T.


9. ∠ATB=120°.
Question 8.

Draw a pair of tangents to a circle of radius 13.2 cm which are inclined to each other at an angle of 70°.
Answer:

Sum of angles between the tangents and corresponding radius is 180°.


As the angle between the tangents is 70° so the angle between the radius will be (180° - 70°) =110°.


Steps of Construction:


1. Draw a circle with center O and radius 2.4cm.


2. Draw a radius OA of this circle.



3. Draw another radius OB which is 110° to OA.



4. Extend OA such that OA=AP.


5. Draw a perpendicular bisector of OP.



6. Extend OB such that OB=OQ.


7. Draw a perpendicular bisector of OQ.



8. The two perpendicular bisectors meet at T.


9. ∠ATB=70°.
Question 9.

Draw a circle of radius 3 cm. Take a point P at a distance of 5 cm from the center O of the circle from the point P, draw two tangents to the circle.
Answer:

Steps of construction:


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


2. Draw OP=5cm.



3. Draw a perpendicular bisector of OP. It intersects with OP at M.



4. With M as center and MP as radius, draw a circle.



5. The circles intersect at Q and R.


6. Join PQ and PR.



7. PQ and PR are the required tangents.
Question 10.

The centers of two circles of radii 3 cm and 4 cm are 8 cm apart. How many common tangents can be drawn to the two circles? Also, construct two direct common tangents to the circles.
Answer:


Steps of Construction:”

1. Draw a line segment AB of length 8cm.


2. With A as center, draw a circle of radius 3 cm.


3. With B as center, draw a circle of radius 4 cm.



4. With B as center, draw another circle of radius (4-3) =1cm.



5. Draw a perpendicular bisector of AB. It intersects with AB at X.



6. With X as center and AX as radius draw a circle.



7. This circle intersects with circle of radius 1cm at P and Q.


8. Join B to P. Extend BP to R, a point on the outer circle.


9. Join B to Q. Extend BQ to S, a point on the outer circle.



10. From A, draw a line parallel to BR and another line parallel to BS.


11. These lines intersects with the circle at K and L.



12. Join K to R and L to S.


13. KR and LS are the required direct common tangents.
Question 11.

The radii of two circles are 1.7 cm and 2.8 cm, and their centers are 6 cm apart. Construct a transverse common tangent to the two circles.
Answer:


Steps of Construction:”

1. Draw a line segment AB of length 6cm.


2. With A as center, draw a circle of radius 1.7 cm.


3. With B as center, draw a circle of radius 2.8 cm.



4. With B as center, draw another circle of radius (1.7+2.8) =4.5cm.



5. Draw a perpendicular bisector of AB. It intersects with AB at M.



6. With M as center and AM as radius draw a semicircle.



7. This circle intersects with circle of radius 5cm at S.


8. Join B to S. BS cuts the inner circle at K.



9. From A, draw a line parallel to BS.


10. This line intersects with the circle at L.



11. Join K to L.


12. KL is the required transverse common tangent.
Exercise 14.2
Question 1.

Write the truth value (T/F) of each of the following statements. If possible, give reason for your answer.

(i) In an equilateral triangle, the incircle and the circumcircle can be constructed with the same centre.

(ii) The incircle of a triangle touches all sides of the triangle.

(iii) In an obtuse-angled triangle, the circumcenter lies on one of the sides of the triangle.

(iv) In an acute-angled triangle, the circumcentre lies inside the triangle.

(v) The incircle of a triangle is constructed by locating the point of intersection of perpendicular bisectors of any two sides of the triangle.
Answer:

(i) True. In an equilateral triangle, circumcenter and incenter coincide and hence both the circles can be drawn using the same center.


(ii) True, because while constructing an incircle, a perpendicular is dropped on a side and length of the perpendicular is taken as the radius.


(iii) False. Only in a right-angled triangle does the circumcenter lie on the hypotenuse.


(iv) True. The perpendicular bisectors of the sides intersect inside the triangle.


(v) False. Incenter is the intersection of the angle bisectors of two of the angles.
Question 2.

Draw incircle of an equilateral triangle of side 4.6 cm. Do its circumcentre and incentre coincide with each other? Give reason for your answer
Answer:

Steps of Construction:


1. Draw an equilateral triangle of side 4.6cm



2. Draw angle bisector of ∠A.



3. Draw angle bisector of ∠B.



4. They intersect each other at P.


5. From P, drop a perpendicular on the line AB.



6. With P as center and length of perpendicular as the radius, draw a circle.



This circle is the incircle for the equilateral triangle.


Yes. The incenter and circumcenter of the equilateral triangle coincide with each other. This is because the perpendicular dropped from P on AP is also the perpendicular bisector of AB.
Question 3.

Draw incircle of a triangle ABC where AB = 4.6 cm, AC = 4.2 cm and ∠A = 90°.
Answer:

Steps of Construction:


1. Draw a right-angled triangle of given dimensions.



2. Draw angle bisector of ∠B.



3. Draw angle bisector of ∠A.



4. They intersect each other at P.


5. From P, drop a perpendicular on the line AB.



6. With P as center and length of perpendicular as the radius, draw a circle.
Question 4.

Draw circumcircle of a triangle whose sides are 10.5 cm, 12.7 cm and 13 cm respectively. Also give reason as to why its circumcentre is located on the side of length 13 cm.
Answer:

Let the triangle be ABC,


1. Draw a triangle ABC of given dimensions.



2. Now, draw perpendicular bisector of side AB.



3. Now, draw perpendicular bisector of BC.



4. Now, draw circle with the point of intersection of perpendiculars as center and passing through all vertices of the triangle.



Here we can see, the circle is touching all the vertices of the triangle. Hence it is the circumcircle.


Since the triangle is not a right angled triangle so the circumcenter is not lying on any side.
Question 5.

Where should the circumcentre of a triangle be located if sides of the triangle are 5 cm, 4.5 cm and 7 cm? Verify your answer by actual construction. Also, draw circumcircle to the given triangle.
Answer:

The circumcenter of this triangle should lie outside the triangle as it is an obtuse angled triangle.


Steps of construction:


1. Construct a triangle ABC of given dimensions.



2. Draw a perpendicular bisector of AB.



3. Draw a perpendicular bisector of BC



4. These perpendicular bisectors intersect at O.


5. With O as center and OA as radius, draw a circle.



6. This circle is the circumcircle of the given triangle.
Question 6.

Construct ΔABC where AB = 6 cm, BC = 4 cm and ∠B = 120°. Also draw incircle of the triangle.
Answer:

Steps of Construction:


1. Construct a triangle of given dimensions.



2. Draw angle bisector of ∠B.



3. Draw angle bisector of ∠A.



4. They intersect each other at P.


5. From P, drop a perpendicular on the line AB.



6. With P as center and length of perpendicular as the radius, draw a circle.