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**SHORT ANSWER TYPE QUESTIONS**

**Q1. Draw a circle of diameter 6.4 cm. Then draw two tangents to the circle from a point P at a distance 6.4 cm from the centre of the circle.**

**Sol. Steps of construction:**

**I. **Draw a circle with centre O and radius = 6.4/2 cm or 3.2 cm.

II. Mark a point P outside the circle such that OP = 6.4 cm.

III. Join OP.

IV. Bisect OP such that its mid point is at M.

V. With centre M and radius OM, draw a circle intersecting the given circle at A and B.

VI. Join PA and PB. Thus, PA and PB are the two tangents to the given circle.

**Q2. Draw a circle of radius 3.4 cm. Draw two tangents to it inclined at an angle of 60Â° to each other:**

**Sol. Steps of construction: **

I. Draw a circle with centre O and radius as 3.4 cm.

II. Draw two radii OA and OB such that âˆ AOB = 120Â°.

III. Draw perpendiculars at A and B such that these perpendiculars meet at P.

Obviously, âˆ APB = 60Â°. [Using Angle sum property of a quadrilateral]

IV. Thus, PA and PB are the required tangents to the given circle.

**Q3. Draw Î” ABC in which AB = 3.8 cm, âˆ B = 60Â° and median AD = 3.6 cm. Draw another triangle ABâ€™Câ€™ similar to the first such that AB = (4/3)AB.**

**Sol. Steps of construction: **

I. Draw AB = 3.8 cm.

II. Construct âˆ ABY = 60Â°.

III. With centre A and radius as 3.6 cm mark a ray to intersect BY at D.

IV. With centre D and radius BD, mark an arc to intersect BY at C.

V. Join CA. Thus, ABC is a triangle.

VI. Draw a ray AX, such that âˆ BAX is an acute angle.

VII. Mark 4 points X_{1}, X_{2}, X_{3} and X_{4} such that AX_{1} = X_{1}X_{2} = X_{2} X_{3} = X_{3} X_{4}.

VIII. Join X_{3}B.

IX. Through X4 draw X4Bâ€™ y X3B X. Through Bâ€™ draw Bâ€™ Câ€™ â•‘ BC where Câ€² lies on AC (produced).

Thus, Î”Câ€™AB is the required triangle.

**Q4. Draw an equilateral triangle of height 3.6 cm. Draw another triangle similar to it such that its side is 2/3 of the side of the first.**

**Sol. Steps at construction: **

I. Draw a line segment RS.

II. Mark a point Y on it.

III. Through Y, draw YZ âŠ¥ RS

IV. Mark a point A on YZ such that YA = 3.6 cm

V. At A draw âˆ YAB = 30Â° such that the point B is on RS.

VI. With centre A and radius = AB, mark a point C on RS.

VII. Join AC.

VIII. Draw a ray BX such that âˆ CBX is an acute angle.

IX. Mark three points X_{1}, X_{2}, X_{3} such that AX_{1} = X_{1}X_{2} = X_{2} X_{3}.

X. Join X_{3} and C.

XI. Through X_{2} draw X_{2}Câ€² â•‘ X_{3}C.

XII. Through Câ€² draw Câ€² Aâ€² â•‘ CA.

Thus, âˆ†Aâ€² BCâ€² is the required triangle.

**Q5. Draw an isosceles âˆ† ABC, in which AB = AC = 5.6 cm and âˆ ABC = 60Â°. Draw another âˆ† ABâ€² Câ€² similar to âˆ†ABC such that ABâ€² = (2/3) AB.**

**Sol. Steps of Construction: **

I. Draw a ray BD.

II. Through B, draw another ray BE such that âˆ DBE = 60Â°.

III. Cut off BA = 5.6 cm.

IV. With A as centre and radius 6 cm, mark an arc intersecting BD at C.

V. Join A and C to get Î”ABC.

VI. Draw a ray BX such that âˆ CBX is an acute angle.

VII. Mark three point X_{1}, X_{2} and X_{3} such that BX_{1} = X_{1}X_{2} = X_{2}X_{3}.

VIII. Join X_{3} and C.

IX. Through X_{2} draw X_{2}Câ€² â•‘ X_{3}C

X. Through Câ€² , draw Câ€² Aâ€² â•‘ CA

Thus, Î” Aâ€² BCâ€² is the required triangle.

**Q6. Construct an isosceles triangle whose base is 9 cm and altitude is 5 cm. Then construct another triangle whose sides are 3/4 of the corresponding sides of the first isosceles triangle.**

**Sol. Steps of construction: **

I. Construct a Î”ABC such that AB = AC, BC = 9 cm and altitude AD = 5 cm.

II. Through B, draw a ray BX such that âˆ CBX is an acute angle.

III. Mark 4 equal points X_{1}, X_{2}, X_{3} and X_{4} on BX. such that BX_{1} = X_{1} X_{2} = X_{2} X_{3} = X_{3} X_{4}

IV. Join X_{4} and C.

V. Through X_{3}, draw X_{3}Câ€² â•‘ X_{4}C, intersecting BC in Câ€².

VI. Through Câ€² , draw Câ€² Aâ€² â•‘ CA, intersecting AB in Aâ€².

Thus, Î”Aâ€²BCâ€² is the required triangle.

**Q7. Draw a line segment AB of length 7 cm. Taking A as centre draw a circle of radius 3 cm and taking B as centre, draw another circle of radius 2.5 cm. Construct tangents to each circle from the centre of the other circle.**

**Sol. Steps of construction: **

I. Draw a line segment AB = 7 cm

II. With centre A and radius 3 cm, draw a circle.

III. With centre B and radius 2.5 cm, draw another circle.

IV. Bisect AB and let M be the mid point of AB.

V. With centre M and radius AM, draw a circle intersecting the two circles in P,Q and R,S.

VI. Join AP, AQ, BR and BS.

Thus, AP, AQ, BR and BS are required tangents.

**Q8. Construct a Î” ABC in which BC = 6.5 cm, AB = 4.5 cm and âˆ ABC = 60Â°. Construct a triangle similar to this triangle whose sides are 3/4 of the corresponding sides of the Î”ABC.**

**Sol. Steps of construction: **

I. Construct the Î”ABC such that AB = 4.5 cm, âˆ B = 60Â° and BC = 6.5 cm.

II. Construct an acute angle âˆ BAX.

III. Mark 4 points X_{1}, X_{2}, X_{3} and X_{4} on AX such that AX_{1} = X_{1}X_{2} = X_{2}X_{3} =X_{3}X_{4}.

IV. Join X_{4} and B.

V. Draw X_{3} Bâ€² â•‘ BC, meeting AC at Câ€² .

Thus, Î”Câ€²ABâ€² is the required Î”.

**Q9. Draw a right triangle in which sides (other than hypotenuse) are of lenghts 8 cm and 6 cm. Then construct another triangle whose sides are 3/4 times the corresponding sides of the first triangle.**

**Sol. Steps of construction:**

I. Draw a Î”ABC such that AB = 8 cm, âˆ B = 90Â° and BC = 6 cm.

II. Construct an acute angle âˆ BAX.

III. Mark 4 points X_{1}, X_{2}, X_{3} and X_{4} on AX such that AX_{1} = X_{1}X_{2}, = X_{2}X_{3} = X_{3}X_{4}.

IV. Join X_{4} and B.

V. Draw X_{3}Bâ€² â•‘ X4B.

VI. Draw Bâ€² Câ€² â•‘ BC.

Thus, Î”ABâ€²Câ€² is the required rt Î”.

**Q10. Construct a Î”ABC in which BC = 5 cm, CA = 6 cm and AB = 7 cm. Construct a Î”Aâ€² BCâ€² similar to Î”ABC, each of whose sides are 7/5 times the corresponding sides of Î”ABC.**

**Sol. Steps of construction: **

I. Construct Î”ABC such that: BC = 5 cm, CA = 6 cm and AB = 7 cm.

II. Draw a ray BX such that âˆ CBX is an acute angle.

III. Mark 7 points X_{1}, X_{2}, .............. X_{7} such that: BX_{1} = X_{1}X_{2}, = X_{2}X_{3} = X_{3}X_{4} = X_{4}X_{5} = X_{5}X_{6} = X_{6}X_{7}

IV. Join X_{7} and C.

V. Draw a line through X_{5} parallel to X_{7}C to meet BC extended at Câ€² .

VI. Through Câ€² , draw a line parallel to CA to meet BA extended at Aâ€² .

Thus, Î”Aâ€²BCâ€² is the required triangle.

**Q11. Construct a triangle with sides 4 cm, 5 cm and 7 cm. Then construct a triangle similar to it whose sides are 2/3 of the corresponding sides of the given triangle.**

**Sol. Steps of construction: **

I. Construct the Î”ABC such that BC = 7 cm, CA = 5 cm and BA = 4 cm.

II. Draw a ray BX such that âˆ CBX is an acute angle.

III. Mark three points X_{1}, X_{2} and X_{3} on BX such that: BX_{1} = X_{1} X_{2} = X_{2} X_{3}

IV. Join X_{3} and C.

V. Draw X_{2}Câ€² â•‘ X_{3}C.

VI. Draw Câ€² Aâ€² â•‘ CA

Thus, Î”Aâ€² BCâ€² is the required triangle.

**Q12. Construct a Î”ABC in which AB = 6.5 cm, âˆ B = 60Â° and BC = 5.5 cm. Also construct a triangle ABâ€²Câ€² similar to Î”ABC whose each side is 3/2 times the corresponding side of the Î”ABC.**

**Sol. Steps of construction:**

I. Construct a Î”ABC such that AB = 6.5 cm, âˆ B = 60Â° and BC = 5.5 cm.

II. Draw a ray AX making an acute angle âˆ BAX.

III. Mark three points X_{1}, X_{2}, X_{3} on the ray AX such that AX_{1} = X_{1} X_{2} = X_{2} X_{3}

IV. Join X_{2} and B.

V. Draw X_{3}Bâ€² â•‘ X_{2}B such that Bâ€² is a point on extended AB.

VI. Join Bâ€² Câ€² y BC such that Câ€² is a point on AC (extended).

Thus, Î”Câ€² ABâ€² is the required triangle.

**Q13. Draw a Î”ABC with side BC = 6 cm, AB = 5 cm and âˆ ABC = 60Â°. Construct Î”ABâ€² Câ€² similar to Î”ABC such that sides of Î”ABâ€²Câ€² are 3/4 of the corresponding sides of Î”ABC.**

**Sol. Steps of construction: **

I. Construct the given Î”ABC.

II. Draw a ray AX such that âˆ BAC is an acute angle.

III. Mark 4 points X_{1}, X_{2}, X_{3} and X_{4} on such that AX_{1} = X_{1} X_{2} = X_{2} X_{3} = X_{3} X_{4}.

IV. Join X_{4} B.

V. Draw X_{3}Bâ€² â•‘ X_{4}B

VI. Through Bâ€² draw Bâ€² Câ€² â•‘ BC.

Thus, Î”Bâ€² ACâ€² is the required triangle.

**Q14. Draw a circle of radius 3 cm. From a point P, 6 cm away from its centre, construct a pair of tangents to the circle. Measure the lengths of the tangents.**

**Sol. Steps of construction:**

I. Draw the given circle such that its centre is at O and radius = 3 cm.

II. Mark a point P such that OP = 6 cm.

III. Bisect OP. Let M be the mid point of OP.

IV. Taking M as centre and OM as radius draw a circle intersecting the given circle at A and B.

V. Join PA and PB.

Thus, PA and PB are the required tangents to the given circle.

**Q15. Construct a triangle whose perimeter is 13.5 cm and the ratio of the three sides is 2 : 3 : 4.**

**Sol. Steps of construction: **

I. Draw a line PQ = 13.5 cm

II. At P, draw a ray PR making a convenient acute angle â€“QPR with PQ.

III. On PR mark (2 + 3 + 4), 9 points at equal distances.

IV. Join Q and the mark 9.

V. Through the points 2 and 5 draw lines 2-A and 5-B parallel to 9-Q. Let these lines meet PQ at A and B respectively.

VI. With A as centre and radius = AP, draw an arc.

VII. With B as centre and radius = BQ, draw another arc which intersects the arc of step

VI at C.

VIII. Join CA and CB.

ABC is the required triangle.

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