# Question-1

**Draw a circle of radius 3 cm. From a point 10 cm away from its centre. Construct the pair of tangents to the circle.**

**Solution:**

**Given:**A circle with centre O and radius 3 cm.

**Required:**To construct the pair of tangents.

**Steps of Construction:**

(i) Draw a circle of radius 3 cm.

(ii) Take an external point P which is 10 cm away from its centre. Join OP.

(i) Bisect the line segment OP = 10 cm. Let the point of bisection be M.

(ii) Taking M as centre and OM as radius, draw a circle. Let it intersect the given circle at the points Q and R.

(iii) Join PQ and PR.

These are the required tangents.

# Question-2

**Construct a triangle ABC whose sides are 7.5 cm, 7 cm and 6.5 cm. Construct another triangle similar to Î” ABC and with sides of the corresponding sides of triangle ABC.**

**Solution:**

**Given:**Î” ABC, AB = 7.5 cm, BC = 7 cm and CA = 6.5 cm.

**Required:**To construct a Î” Aâ€™BCâ€™ in which Aâ€™B = AB, Aâ€™Câ€™ = AC and BCâ€™ = BC.

**Steps of construction:**

(i) Divide the base BC into three equal parts. Let Câ€™ be the point on BC such that BCâ€™ = BC.

**Steps of construction**

1. Draw a line segment BC = 7 cm, AB = 7.5 cm and CA = 6.5 cm.

2. Below BC, make an acute angle âˆ CBP

3. Divide the base BC into three equal parts. Let Câ€™ be the point on BC such that BCâ€™ = BC.

4. Along BP, mark off three points X_{1}, X_{2}, X_{3} such that XX_{1} = X_{1X}_{2} = X_{2X}_{3}

5. Join X_{3}C

6. Draw a line Câ€™Aâ€™ || CA intersecting BA at Aâ€™.

Then Aâ€™BCâ€™ is the required triangle.

# Question-3

**Construct a triangle similar to a given triangle with sides 5 cm, 12 cm and 13 cm and whose sides are of the corresponding sides of the given triangle.****Solution:**

**Given:**Î” ABC, AB = 5 cm, BC = 12 cm and CA = 13 cm.

**Required:**To construct a Î” Aâ€™BCâ€™ in which Aâ€™B =AB, Aâ€™Câ€™ =AC and BCâ€™ =BC.

**Steps of construction:**

1. Draw a line segment BC = 12 cm

2. With B as centre and with radius 5 cm, draw an arc.

3. With C as centre and with radius 13 cm, draw another arc, intersecting the previously drawn arc at A.

4. Join AB and AC. Then, Î” ABC is the required triangle.

5. Below BC, make an acute angle âˆ CBP.

6. Along BP, mark off seven points X_{1}, X_{2}, X_{3}â€¦..X_{7} such that XX_{1} = X_{1X}_{2}â€¦â€¦ X_{6}X_{7}

7. Join X_{5} to C and draw a line through X_{3} parallel to X_{5} C, intersecting the extended line segment BC at Câ€™.

8. Draw a line through Câ€™ parallel to CA intersecting the line segment BA at Aâ€™. Then Aâ€™BCâ€™ is the required triangle.

# Question-4

**Construct a triangle similar to a given triangle with sides 6 cm, 7 cm and 8 cm and whose sides are th of the corresponding sides of the given triangle.**

**Solution:**

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**Steps of construction:**

1. Draw a line segment BC = 7 cm

2. With B as centre and with radius 6 cm, draw an arc.

3. With C as centre and with radius 8 cm, draw another arc, intersecting the previously drawn arc at A.

4. Join AB and AC. Then, Î” ABC is the required triangle.

5. Below BC, make an acute angle âˆ CBX.

6. Along BX, mark off seven points X

_{1}, X

_{2}, X

_{3}â€¦..X

_{7}such that XX

_{1}= X

_{1X}

_{2}â€¦â€¦ X

_{6}X

_{7}

7. Join X

_{7}to C and draw a line through X

_{5}parallel to X

_{7}C, intersecting the extended line segment BC at Câ€™.

8. Draw a line through Câ€™ parallel to CA intersecting the line segment BA at Aâ€™. Then Aâ€™BCâ€™ is the required triangle.

# Question-5

**Divide a line segment of 7 cm length externally in the ratio of 3 : 5.****Solution:**

**Given:**AB is a line segment of 7 cm length.

**Required: **To divide a line segment of 7 cm length externally in the ratio of 3 : 5.

** **

**Steps of Construction:**

1. Draw the line segment AB = 7 cm.

2. Draw ray BX making an acute âˆ ABX .

3. Along BX, mark off five points B_{1,}, B_{2}, B_{3,}, B_{4} and B_{5.} Join B_{2 }to A.

4. Through B_{5} draw B_{5}P || B_{2}A , intersecting BA produced at P.

5. The point P so obtained is the required point which divides AB externally in the ratio

3 : 5.

**Proof: **In Î”s ABB_{2} and PBB_{5},

B_{5}P || B_{2}A â‡’ ABB_{2} âˆ¼ PBB_{5}

âˆ´ =(Property of similarty).

â‡’

# Question-6

**Construct a triangle similar to a given Î” ABC such that each of its sides is rd of the corresponding sides of the Î”ABC. Given AB = 4.2 cm, BC = 5 cm and AC = 6.2 cm.**

**Solution:**

**Given:**In Î” ABC, AB = 4.2 cm, BC = 5 cm and AC = 6.2 cm.

**Required**: To construct Î” ABâ€™Câ€™ such that each of its sides is rd of the corresponding sides of the Î” ABC.

__ __

__Steps of Construction:__

1. Draw a line segment AB = 4.2 cm.

2. With A as centre and radius = AC = 6.2 cm, draw an arc.

3. With B as centre and radius = BC = 5 cm, draw another arc, intersecting the previous arc at C.

4. Join AC and BC to obtain Î” ABC.

5. Below AB, make an acute angle âˆ BAX.

6. Along AX, mark off three points A_{1} , A_{2} , A_{3 }such that AA_{1} = A_{1}A_{2} = A_{2}A_{3}

7. Join A_{3}B.

8. Draw A_{2}Bâ€™ || A_{3}B, meeting AB at Bâ€™.

9. From Bâ€™, draw Bâ€™Câ€™ || BC meeting AC at Câ€™.

ABâ€™Câ€™ is the required Î” .

**Proof**: Since Bâ€™Câ€™ || BC , Î” ABC âˆ¼ Î” ABâ€™Câ€™.

Bâ€™Câ€™/BC = ACâ€™/AC = ABâ€™/AB = 2/3.

# Question-7

**Construct a triangle similar to a Î” XYZ with its sides equal to ()th of the corresponding sides of Î”XYZ. It is given that XY = 6 cm, XZ = 5 cm and ZY = 4 cm. Write the steps of construction.****Solution:**

**Given:**

**D**XYZ in which XY = 6 cm, XZ = 5 cm and ZY = 4 cm.

**Required:**To construct a Î” XY'Z' in which XYâ€™ = (3/4)XY, Yâ€™Zâ€™ = (3/4)ZY and

XZâ€™ = (3/4)XZ.

**Steps of construction: **

(i) Draw a ray XP.

(ii) Construct a Î” XYZ in which XY = 6 cm, XZ = 5 cm and ZY = 4 cm.

(iii) Draw any ray XP inclined at certain angle with X.

(iv) Starting from X, cut off seven equal line â€“ segment XX_{1}, X_{1}X_{2}, X_{2}X_{3}, X_{3}X_{4} on XQ.

(v) Join YX_{4} and draw a line â€“ segment X_{3}Yâ€™ parallel to X_{4}Y to intersect XP at Yâ€™_{. }

Draw a line Yâ€™Zâ€™ parallel to YZ which intersects XP in Yâ€™

Then XYâ€™Zâ€™ is the required quadrilateral.

# Question-8

**Draw a Î” ABC in which AB = 5 cm, BC = 4.6 cm, and AC = 5.8 cm. Construct a triangle similar to Î” ABC such that each of its sides is 2/3**

^{rd}of the corresponding sides of Î” ABC.**Solution:**

**Given**: In Î” ABC, in which AB = 5 cm, BC = 4.6 cm, and AC = 5.8 cm.

**Required:**To construct a triangle similar to Î” ABC such that each of its sides is two-third of the corresponding sides of Î” ABC.

**Steps of Construction:**

(i) Draw BC = 4.6 cm.

(ii) With B as centre and radius equal to 5 cm draw an arc and with C as centre and radius equal

to 5.8 cm draw another arc to cut the previous arc at A.

(iii) Join AB and AC.

(iv) Make an acute angle âˆ CBE.

(v) Set off three equal distances along BE at B_{1, }B_{2 } and B_{3}.

(vi) Join B_{3}C.

(vii) From B_{2} draw B_{2}Câ€™ || B_{3}C, meeting BC at Câ€™.

(viii) Join ACâ€™.

Then, ABCâ€™ is the required triangle.