# Frank Solutions for Chapter 18 Construction Class 10 ICSE Mathematics

**Exercise 18**

**1. Draw a circle of radius 4 cm and take a point P on its circumference. Construct a tangent to the circle at P.**

**Answer**

**Steps for constructions:**

- Draw a circle of radius 4 cm with centre O.
- Then join the center O to the given point P.
- On the give point P, draw a perpendicular to OP.
- So, PQ is the required tangent.

**2. Draw a circle of radius 4.5 cm. Take a point P on its circumference. Construct a tangent to the circle at P without using the center.**

**Answer**

**Steps for constructions:**

- Draw a circle of radius 4.5 cm.
- Take any point P draw a chord PX.
- Take any point Y on the circle and join PX and PY.
- Then, at P draw ∠YPZ equal to ∠XYP.
- Therefore, PZ is the required tangent.

**3. Draw a circle with center O and radius 2.5 cm. Take a point P at distance of 6 cm from the center. Using ruler and compasses only construct the tangent to the circle from the point.**

**Answer**

**Steps for constructions:**

- Draw a circle of radius 2.5 cm with centre O.
- Then join the center O to the given point P which is 6 cm away from O.
- Now draw a perpendicular bisector of OP. let us assume Z be the mid – point OP.
- With Z as center and radius OZ, draw a circle cutting the first circle at X and Y.
- Now join PX and PY.
- Therefore, PX and PY are the required tangents.

**4. Draw a circle with center O and radius 3 cm. Take a point P outside the circle. Draw tangents to the circle from P without using the center and using only ruler and compasses.**

**Answer**

**Steps for constructions:**

- Draw a circle of radius 3 cm.
- P is the given point, then draw PXY a secant to the given circle.
- Then, draw a perpendicular bisector of PY and let us assume Q be the mid-point of PY.
- Now, Q as the center and QP as radius, draw a semi-circle on PY.
- At X, draw a perpendicular to PY. Let this perpendicular meet the semi-circle at D.
- With P as center and PL as radius, cut off two arcs on the given circle at Z and R.
- Then, join PZ and PR.
- Therefore, PZ and PS are the required tangents.

**5.** **Using ruler and compass only, draw tangents to a circle of radius 3 cm from a point 5 cm from the center. What is the length of each of them?**

**Answer**

**Steps for constructions:**

- Draw a circle of radius 3 cm with center O.
- Now join the center S to the point P which is 5 cm away from S.
- Then, draw a perpendicular bisector of SP. Let us assume O be the mid-point of SP.
- Take O as center and radius OS, draw a circle cutting the first circle at Q and R.
- Now join PQ and PR.
- Therefore, PQ and PB are the required tangents.
- Then the measure of PQ and PB are equal to 4 cm.

**6. ****Draw two circles of radii 2.5 cm and 3.5 cm respectively so that their centers are 8 cm apart. Draw direct common tangents to the circle.**

**Answer**

**Steps for constructions:**

- First draw a line AB = 8 cm.
- At A, draw a circle of radius 3.5 cm.
- At B, draw a circle of radius 2.5 cm.
- At A, draw a third circle concentric to the bigger circle and radius, = 3.5 – 2.5 = 1 cm
- Then draw a perpendicular bisector of AB, let H be the mid-point of AB.
- So, take H as center and AH as radii, draw a fourth circle. Mark as G and I where the third and fourth circles intersect each other.
- Now, join AG and AI and external line to meet the bigger circle at E and C.
- Then join BG and BI.
- On BG and BI, draw perpendicular to meet the smaller circle at F and D.
- Now join EF and CD.
- Therefore, EF and CD are the required tangents.

**7.**

**Draw two circles of radii 3.5 cm and 2 cm respectively so that their centers are 6 cm apart. Draw direct common tangents to the circle and show that they are equal in length.**

**Answer**

- First draw a line AB = 6 cm.
- At A, draw a circle of radius 3.5 cm.
- At B, draw a circle of radius 2 cm.
- At A, draw a third circle concentric to the bigger circle and radius, = 3.5 – 2 = 1.5 cm
- Then draw a perpendicular bisector of AB, let H be the mid-point of AB.
- So, take H as center and AH as radii, draw a fourth circle. Mark as G and I where the third and fourth circles intersect each other.
- Now, join AG and AI and external line to meet the bigger circle at E and C.
- Then join BG and BI.
- On BG and BI, draw perpendicular to meet the smaller circle at F and D.
- Now join EF and CD.

**… [tangents to a circle from same point]**

**8. Draw two circles of radii 3 cm and 3.5 cm, their centers being 8 cm apart. Construct a transverse common tangent and measure its length.**

**Answer**

**Steps for constructions:**

- First draw a line AB = 8 cm.
- At A, draw a circle of radius 3 cm.
- At B, draw a circle of radius 3.5 cm.
- At A, draw a third circle concentric to the smaller circle and radius,= 3.5 + 3 = 6.5 cm
- Then draw a perpendicular bisector of AB, let H be the mid-point of AB.
- So, take H as center and AH as radii, draw a fourth circle. Mark as G and I where the third and fourth circles intersect each other.
- Now, join AG and AI to meet the smaller circle at E and C.
- Then join BG and BI.
- On BG and BI, draw perpendicular to meet the bigger circle at F and D.
- Now join EF and CD.

Therefore, EF and CD are the required tangents both are equal to 8 cm.

**9. Draw two circles with radii 2.5 cm and 4 cm and with their centers 7 cm apart. Draw a direct common tangent and a transverse common tangent. Calculate the length of the direct common tangent.**

**(i) Transverse common tangent**

**(ii) Direct common tangent**

**Answer**

**(i) Steps for constructions:**

- First draw a line AB = 7 cm.
- At A, draw a circle of radius 2.5 cm.
- At B, draw a circle of radius 4 cm.
- At A, draw a third circle concentric to the smaller circle and radius, = 2.5 + 4 = 6.5 cm
- Then draw a perpendicular bisector of AB, let H be the mid-point of AB.
- Now, join AG and AI to meet the smaller circle at E and C.
- Then join BG and BI.
- On BG and BI, draw perpendicular to meet the bigger circle at F and D.
- Now join EF and CD.

Therefore, EF and CD are the required tangents.

**(ii)**

**Steps for constructions:**

- First draw a line AB = 7 cm.
- At A, draw a circle of radius 4 cm.
- At B, draw a circle of radius 2.5 cm.
- At A, draw a third circle concentric to the bigger circle and radius, = 4 – 2.5 = 1.5 cm
- Then draw a perpendicular bisector of AB, let H be the mid-point of AB.
- Now, join AG and AI and external line to meet the bigger circle at E and C. Then join BG and BI.
- On BG and BI, draw perpendicular to meet the smaller circle at F and D.
- Now join EF and CD.

Therefore, EF and CD are the required tangents both are equal to 7 cm.

**10. Construct Î”ABC in which AB = 5 cm, BC = 4.5 cm and ****∠****ABC = 60 ^{o}. Construct a circle to circumcircle of Î”ABC.**

**Answer**

**Steps for constructions:**

- First draw a line AB = 6 cm.
- At B, draw an arc making an angle of 60
^{o}with AB. - Then on the arc cut BC = 4.5 cm.
- Now join AC.
- Then draw a perpendicular bisector of AB and BC, which meet at O.
- Take O as center and radius equal to the distance between O and the vertex of the triangle, draw a circle to pass through all the three vertices of the triangle.
- Therefore, above circle is the required circle with radius of 2.8 cm.

**11. Using ruler and compasses only, construct a triangle ABC in which AB = 5cm, BC = 6 cm and CA = 4.5 cm. Construct a circle passing through A, B and C.**

**Answer**

**Steps for constructions:**

- First draw a line AB = 5 cm.
- At B, draw an arc making an angle of 6 cm.
- Then on the arc cut AC = 4.5 cm.
- Now join AC and BC.
- Then draw a perpendicular bisector of AB and BC, which meet at O.
- Take O as center and radius equal to the distance between O and the vertex of the triangle, draw a circle to pass through all the three vertices of the triangle.
- Therefore, above circle is the required circle.

**12. Using ruler and compasses only, construct and equilateral triangle with side 4.5 cm. Draw a circumcircle of this triangle and measure its radius.**

**Answer**

**Steps for constructions:**

- First draw a line AB = 4.5 cm.
- At B, draw an arc making an angle of 60
^{o}with AB and length BC = 4.5 cm. - Then on the arc cut AC = 4.5 cm.
- Now join AC.
- Then draw a perpendicular bisector of AB and BC, which meet at O.
- Take O as center and radius equal to the distance between O and the vertex of the triangle, draw a circle to pass through all the three vertices of the triangle.
- Therefore, above circle is the required circle with radius of 2.6 cm.

**13. Using ruler compasses only, construct Î”ABC in which BC = 7.5 cm, ****∠****ABC = 60 ^{o}and AC – AB = 1.5 cm. Inscribe a circle in the Î”ABC and measure its radius.**

**Answer**

**Steps for constructions:**

- First draw a line BC = 7.5 cm.
- At B, draw an arc making an angle of 60
^{o}with BC. - Then on the arc cut AC = AB + 1.5 cm = 7.5 + 1.5 = 9 cm cutting the previous arc
- Now join AC and AB.
- Then draw an angle bisector of ∠A and ∠B, which meet at O.
- Draw a perpendicular to BC from O and mark it as P.
- With OP as radius draw a circle touching all three sides of the triangle.

Therefore, above circle is the required circle with radius of 2.3 cm.

**14. Draw a circle with radius 3 cm and inscribe an equilateral triangle in it.**

**Answer**

**Steps for constructions:**

- First draw a circle with center O and radius = 3 cm.
- Then draw radii OP and OB such that ∠POQ = 360/3 = 120
^{o}. - Now join PQ and cut off arcs PR and QR equal to PQ.
- Join PR and QR.

Therefore, Î”ABC is the required equilateral triangle.

**15. Draw a circle of radius 2.5 cm and circumscribe a square about it.**

**Answer**

**Steps for constructions:**

- First draw a circle with center M and radius = 2.5 cm.
- Then draw two diameters TU and VW of the circle meeting at centre M.
- Now taking MV as radius cut two arcs from V on both sides of V and repeat same with T, U, and W.
- Then mark new points formed as P, Q, R and S.
- So, join PQ, QR, RS and SP.

Therefore, PQRSC is the required square circumscribing the given circle.

**16. Inscribe a regular hexagon in a circle of radius 3.5 cm. **

**Answer**

**17. Circumscribe a regular pentagon about a circle of radius 3 cm. **

**Answer **

**18. Inscribe a regular pentagon a circle of radius 4 cm. Inscribe a circle in this pentagon and measure its radius. **

**Answer**

**19. Draw a line segment OA, 5 cm long, At O, using ruler and compasses only, construct OB such that, ****∠****AOB = 37.5° construct a circle to touch OA at A and to touch OB at B. **

**Answer**

**20. Draw line segments OA = 4.5 cm, Ob = 3.2 cm such that ****∠****AOB = 45°. Construct a circle touching OA at A and passing through B. **

**Answer**