## Chapter 4 – Geometric Constructions

**Practice set 4.1**

**1. ∆ABC ~ ∆LMN. In ∆ABC, AB = 5.5 cm, BC = 6 cm, CA = 4.5 cm. Construct ∆ABC and DLMN such that \(\large \frac {BC}{MN}\) = \(\large \frac {5}{4}\)**

**1. ∆ABC ~ ∆LMN. In ∆ABC, AB = 5.5 cm, BC = 6 cm, CA = 4.5 cm. Construct ∆ABC and DLMN such that \(\large \frac {BC}{MN}\) = \(\large \frac {5}{4}\)**

**Solution:**

**∆ABC ~ ∆LMN … [Given]**

**∴ \(\large \frac {AB}{LM}\) = \(\large \frac {BC}{MN}\) = \(\large \frac {AC}{LN}\) …(i) [c.s.c.t.]**

** **

**Now,**

**\(\large \frac {BC}{MN}\) = \(\large \frac {5}{4}\) …(ii) [Given]**

**∴ \(\large \frac {AB}{LM}\) = \(\large \frac {BC}{MN}\) = \(\large \frac {AC}{LN}\) = \(\large \frac {5}{4}\) … [from (i) and (ii)]**

** **

**∴ \(\large \frac {AB}{LM}\) = \(\large \frac {5}{4}\) ; \(\large \frac {BC}{MN}\) = \(\large \frac {5}{4}\) ; \(\large \frac {AC}{LN}\) = \(\large \frac {5}{4}\)**

** **

**∴ \(\large \frac {5.5}{LM}\) = \(\large \frac {5}{4}\) ; \(\large \frac {6}{MN}\) = \(\large \frac {5}{4}\) ; \(\large \frac {4.5}{LN}\) = \(\large \frac {5}{4}\)**

** **

**∴ LM = \(\large \frac {5.5\,×\,4}{5}\) ; MN = \(\large \frac {6\,×\,4}{5}\) ; LN = \(\large \frac {4.5\,×\,4}{5}\)**

** **

**∴ LM = 4.4 cm , MN = 4.8 cm , LN = 3.6 cm**

**ROUGH FIGURE**

**∆LMN is the required triangle similar to ∆ABC.**

**2. ∆PQR ~ ∆LTR. In ∆PQR, PQ = 4.2 cm, QR = 5.4 cm, PR = 4.8 cm. Construct ∆PQR and ∆LTR, such that \(\large \frac {PQ}{LT}\) = \(\large \frac {3}{4}\)**

**2. ∆PQR ~ ∆LTR. In ∆PQR, PQ = 4.2 cm, QR = 5.4 cm, PR = 4.8 cm. Construct ∆PQR and ∆LTR, such that \(\large \frac {PQ}{LT}\) = \(\large \frac {3}{4}\)**

**Solution:**

**ROUGH FIGURE**

**∆LTR is the required triangle similar to ∆PQR.**

**3. ∆RST ~ ****∆****XYZ. In ∆RST, RS = 4.5 cm, ∠RST = 40°, ST = 5.7 cm Construct ∆RST and ∆XYZ, such that \(\large \frac {RS}{XY}\) = \(\large \frac {3}{5}\).**

**3. ∆RST ~**

**∆**

**XYZ. In ∆RST, RS = 4.5 cm, ∠RST = 40°, ST = 5.7 cm Construct ∆RST and ∆XYZ, such that \(\large \frac {RS}{XY}\) = \(\large \frac {3}{5}\).**

**Solution:**

**∆RST ~ ∆XYZ … [Given]**

**∠RST ≅ ∠XYZ … [c.a.s.t]**

**∵ ∠RST = 40° …[Given]**

**∴ ∠XYZ = 40° …(i)**

** **

**∴ \(\large \frac {RS}{XY}\) = \(\large \frac {ST}{YZ}\) …(i) [c.s.s.t]**

** **

**Now, **

**\(\large \frac {RS}{XY}\) = \(\large \frac {3}{5}\) …(ii) [Given]**

**∴ \(\large \frac {RS}{XY}\) = \(\large \frac {ST}{YZ}\) = \(\large \frac {3}{5}\) … [from (i) and (ii)]**

** **

**∴ \(\large \frac {RS}{XY}\) = \(\large \frac {3}{5}\) ; \(\large \frac {ST}{YZ}\) = \(\large \frac {3}{5}\)**

** **

**∴ \(\large \frac {4.5}{XY}\) = \(\large \frac {3}{5}\) ; \(\large \frac {5.7}{YZ}\) = \(\large \frac {3}{5}\)**

** **

**∴ XY = \(\large \frac {4.55}{3}\) ; YZ = \(\large \frac {5.75}{3}\)**

** **

**∴ XY = 7.5 cm, YZ = 9.5 cm**

**ROUGH FIGURE**

**∆XYZ is the required triangle similar to ∆RST.**

**4. ∆AMT ~ ∆AHE. In ∆AMT, AM = 6.3 cm, ∠TAM = 50°, AT = 5.6 cm. \(\large \frac {AM}{AH}\) = \(\large \frac {7}{5}\). Construct ∆AHE.**

**4. ∆AMT ~ ∆AHE. In ∆AMT, AM = 6.3 cm, ∠TAM = 50°, AT = 5.6 cm. \(\large \frac {AM}{AH}\) = \(\large \frac {7}{5}\). Construct ∆AHE.**

**Solution:**

**ROUGH FIGURE**

**∆AHE is the required triangle similar to ∆AMT.**

**Practice set 4.2**

**Practice set 4.2**

**1. Construct a tangent to a circle with centre P and radius 3.2 cm at any point M on it.**

**1. Construct a tangent to a circle with centre P and radius 3.2 cm at any point M on it.****Solution:**

**ROUGH FIGURE**

**line l is the required tangent to the circle passing through point M on the circle.**

**2. Draw a circle of radius 2.7 cm. Draw a tangent to the circle at any point on it.**

**2. Draw a circle of radius 2.7 cm. Draw a tangent to the circle at any point on it.****Solution:**

**ROUGH FIGURE**

**line l is the required tangent to the circle passing through point M on the circle.**

**3. Draw a circle of radius 3.6 cm. Draw a tangent to the circle at any point on it without using the centre.**

**3. Draw a circle of radius 3.6 cm. Draw a tangent to the circle at any point on it without using the centre.**

**Solution:**

**ROUGH FIGURE**

**Line m is the required tangent to the circle at point C.**

**4. Draw a circle of radius 3.3 cm Draw a chord PQ of length 6.6 cm. Draw tangents to the circle at points P and Q. Write your observation about the tangents.**

**4. Draw a circle of radius 3.3 cm Draw a chord PQ of length 6.6 cm. Draw tangents to the circle at points P and Q. Write your observation about the tangents.****Solution:**

**Radius = 3.3 cm … [Given]**

**Chord = 6.6 cm … [Given]**

**Chord is twice the radius.**

**Chord PQ is a diameter.**

**ROUGH FIGURE**

**line l and m are the required tangents to the circle at point P and point Q.**

** **

**Tangents at the end points of a diameter are parallel to each other.**

**5. Draw a circle with radius 3.4 cm. Draw a chord MN of length 5.7 cm in it. Construct tangents at point M and N to the circle.**

**5. Draw a circle with radius 3.4 cm. Draw a chord MN of length 5.7 cm in it. Construct tangents at point M and N to the circle.****Solution:**

**ROUGH FIGURE**

**Line MP and line NP are required tangents to the circle at point M and point N respectively.**

**6. Draw a circle with centre P and radius 3.4 cm. Take point Q at a distance 5.5 cm from the centre. Construct tangents to the circle from point Q.**

**6. Draw a circle with centre P and radius 3.4 cm. Take point Q at a distance 5.5 cm from the centre. Construct tangents to the circle from point Q.****Solution:**

**ROUGH FIGURE**

**line MQ and line NQ are the required tangents to the circle from point Q.**

**7. Draw a circle with radius 4.1 cm. Construct tangents to the circle from a point at a distance 7.3 cm from the centre.**

**7. Draw a circle with radius 4.1 cm. Construct tangents to the circle from a point at a distance 7.3 cm from the centre.****Solution:**

**ROUGH FIGURE**

**Line MQ and line NQ are the required tangents to the circle from point Q.**

**Problem set 4**

**Problem set 4****1. Select the correct alternative for each of the following questions.**

**1. Select the correct alternative for each of the following questions.****(1) The number of tangents that can be drawn to a circle at a point on the circle is ______.**

**(A) 3 **

**(B) 2 **

**(C) 1 **

**(D) 0**

**Ans:** Option (C) : 1

**(2) The maximum number of tangents that can be drawn to a circle from a point outside it is ______.**

**(A) 2 **

**(B) 1 **

**(C) one and only one **

**(D) 0**

**Ans:** Option (A) : 2

** **

**(3) If ∆ABC ~ ∆PQR and \(\large \frac {AB}{PQ}\) = \(\large \frac {7}{5}\), then ______.**

**(A) ∆ABC is bigger. **

**(B) ∆PQR is bigger.**

**(C) Both triangles will be equal. **

**(D) Can not be decided.**

**Ans:** Option (A) : ∆ABC is bigger

**2. Draw a circle with centre O and radius 3.5 cm. Take point P at a distance 5.7 cm from the centre. Draw tangents to the circle from point P.**

**2. Draw a circle with centre O and radius 3.5 cm. Take point P at a distance 5.7 cm from the centre. Draw tangents to the circle from point P.****Solution:**

**ROUGH FIGURE**

**3. Draw any circle. Take any point A on it and construct tangent at A without using the centre of the circle.**

**3. Draw any circle. Take any point A on it and construct tangent at A without using the centre of the circle.****Solution:**

**ROUGH FIGURE**

**Line I is the required tangent to the circle at point A.**

**4. Draw a circle of diameter 6.4 cm. Take a point R at a distance equal to its diameter from the centre. Draw tangents from point R.**

**4. Draw a circle of diameter 6.4 cm. Take a point R at a distance equal to its diameter from the centre. Draw tangents from point R.****Solution:**

**ROUGH FIGURE**

**line RA and line RB are the required tangents to the circle at points A and B respectively from point R.**

**5. Draw a circle with centre P. Draw an arc AB of 100° measure. Draw tangents to the circle at point A and point B.**

**5. Draw a circle with centre P. Draw an arc AB of 100° measure. Draw tangents to the circle at point A and point B.****Solution:**

**ROUGH FIGURE**

**Line AQ and line BQ are tangents to the circle at points A and B respectively.**

**6. Draw a circle of radius 3.4 cm and centre E. Take a point F on the circle. Take another point A such that E – F – A and FA = 4.1 cm. Draw tangents to the circle from point A.**

**6. Draw a circle of radius 3.4 cm and centre E. Take a point F on the circle. Take another point A such that E – F – A and FA = 4.1 cm. Draw tangents to the circle from point A.****Solution:**

**ROUGH FIGURE**

**Line AP and line AQ are the required tangents from point A to the circle with centre E.**

**7. ∆ABC ~ ∆LBN. In ∆ABC, AB = 5.1cm, ∠B = 40°, BC=4.8 cm, \(\large \frac {AC}{LN}\) = \(\large \frac {4}{7}\). Construct ∆ABC and ∆LBN.**

**7. ∆ABC ~ ∆LBN. In ∆ABC, AB = 5.1cm, ∠B = 40°, BC=4.8 cm, \(\large \frac {AC}{LN}\) = \(\large \frac {4}{7}\). Construct ∆ABC and ∆LBN.****Solution:**

**ROUGH FIGURE**

**∆LBN is the required triangle similar to ∆ABC.**

**8. Construct ∆PYQ such that, PY = 6.3 cm, YQ = 7.2 cm, PQ = 5.8 cm. If \(\large \frac {YZ}{YQ}\) = \(\large \frac {6}{5}\), then construct ∆XYZ similar to ∆PYQ.**

**8. Construct ∆PYQ such that, PY = 6.3 cm, YQ = 7.2 cm, PQ = 5.8 cm. If \(\large \frac {YZ}{YQ}\) = \(\large \frac {6}{5}\), then construct ∆XYZ similar to ∆PYQ.****Solution:**

**ROUGH FIGURE**

**∆XYZ is the required triangle similar to ∆PYQ.**