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**Exercise 10.4 **

**Question 1: **

**Construct Î”ABC, given mâˆ A = 60:, mâˆ B = 30Â° and AB = 5,8 cm.**

**Answer 1: **

**To construct:** **Î”ABC, given mâˆ A = 60:, mâˆ B = 30Â° and AB = 5,8 cm.**

**Steps of construction: **

**(a) **Draw a line segment AB = 5.8 cm.**(b) **At point A, draw an angle Z YAB = 6ti= with the help of a compass.**(c) **At point B, draw Z XBA = 30" with the help of a compass.**(d) **AY and BX intersect at the point C.

It is the required triangle ABC

**Question 2: **

**Construct Î” PQR if PQ = 5 cm, mâˆ PQR = 105Â° and mâˆ .QRP = 40Â°.**

**Answer 2: **

**Given: **

mâˆ PQR = 105Â° and mâˆ QRP = 40Â°

We know that sum of angles of a triangle is 180Â°.

âˆ´ mâˆ PQR + mâˆ QRP + mâˆ QPR = 180Â°

â‡’ 105Â° + 40Â° + mâˆ QPR= 180Â°

â‡’ 145Â° + mâˆ QPR= 180Â°

â‡’ mâˆ QPR= 180Â° - 145Â°

â‡’ mâˆ QPR = 35Â°

**To construct**: Î”PQR where mâˆ P = 35Â°, mâˆ Q = 105Â° and PQ = 5 cm.

**Steps of construction: **

**(a)** Draw a line segment PQ = 5 cm,**(b) **At point P, draw âˆ XPQ = 35Â° with the help of protractor.**(c)** At point Q, draw âˆ YQP = 105Â° with the help of protractor.**(d)** XP and YQ intersect at point R.

It is the required triangle PQR.

**Question 3: **

**Examine whether you can construct Î”DEF such that EF = 7.2 cm, mâˆ E= 110Â° and mâˆ F = 80Â°. Justify your answer.**

**Answer 3: **

Given: In Î”DEF, mâˆ E = 110Â° and mâˆ T = 80Â°

Using angle sum property of triangle

âˆ D + âˆ E + âˆ F = 180Â°

â‡’âˆ D + 110Â°+ 80Â° = 180Â°

â‡’ âˆ D + 190Â° = 180Â°

â‡’ âˆ D = 180Â° - 190Â° = -10Â°

Which is not possible.

**Exercise 10.5 **

**Question 1: **

**A right angled triangle PQR where mâˆ Q = 9ff, QR= 8 cm and PQ = 10 cm.**

**Answer 1: **

**To construct:** Construct the right angled Î” PQR, where mâˆ Q = 90", QR = 8 cm and PR = 10 cm.

**Steps of construction: **

(a) Draw a line segment QR = 8 cm,

(b) At point Q, draw QX âŠ¥ QR,

(c) Taking R as centre, draw an arc of radius 10 cm.

(d) This arc cuts QX at point P.

(e) foin PQ.

It is the required right-angled triangle PQR,

**Question 2: **

**Construct a right-angled triangle whose hypotenuse is 6 cm long and one the legs is 4 cm long. **

**Answer 2: **

**To construct:** A right-angled triangle DEF where DF = 6 cm and EF = 4 cm

**Steps of construction: **

(a) Draw a line segment EF = 4 cm.

(b) At point Q, draw EX âŠ¥ EF.

(c) Taking F as centre and radius 6 cm, draw an arc. (Hypotenuse)

(d) This arc cuts the EX at point D.

(e) Join DF.

It is the required right-angled triangle DEF.

**Question 3: **

Construct an isosceles right angled triangle ABC, where mâˆ ACB = 90Â° and AC = 6 cm.

**Answer 3: **

**To construct: ** An isosceles right angled triangle ABC where mâˆ C = 90% AC = BC = 6 cm.

**Steps of construction: **

(a) Draw a line segment AC = 6 cm.

(b) At point C, draw XC âŠ¥ CA.

(c) Taking C as centre and radius 6 cm, draw an arc.

(d) This arc cuts CX at point B.

(e) Join BA.

It is the required isosceles right-angled triangle ABC.

**Miscellaneous Questions **

**Questions: **

**Below are given the measures of certain sides and angles of triangles. Identify those which cannot be constructed and say why you cannot construct them. Construct rest of the triangle. **

Triangle | Given measurements | |||

| Î”ABC | mâˆ A | mâˆ B | AB |

2. | Î”PQR |
| mâˆ R | QR = 4.7 cm |

3. | Î”ABC |
| mâˆ B = 50Â° | AC |

4. | Î”LMN | mâˆ L | mâˆ N | LM |

5. | Î”ABC | BC | AB | AC |

| Î”PQR | PQ | QR | PR |

7. | Î”XYZ | XY | YZ | XZ |

3. | Î”DEF | DE | EF | DF |

**Answer 1: **

In Î”ABC, mâˆ A = 85^{o} , mâˆ B = 115^{o} , AB = 5 cm

Construction of Î”ABC is not possible because mâˆ A = 85^{o} + mâˆ B = 200o, and we know that the sum of angles of a triangle should be 180^{o}.

**Answer 2: **

**To construct:** Î”PQR where mâˆ Q = 30^{o} , mâˆ R = 60^{o} and QR = 4.7 cm.

**Steps of construction: **

(a) Draw a line segment QR = 4.7 cm.

(b) At point Q, draw âˆ XQR = 30' with the help of a compass.

(c) At point R, draw âˆ YRQ = 60' with the help of a compass.

(d) QX and RY intersect at point P.

It is the required triangle PQR.

**Answer 3:**

We know that the sum of angles of a triangle is 180Â°.

mâˆ A + mâˆ B + mâˆ C = 180Â°

âˆ´ 70Â° + 50Â° + mâˆ C = 180Â°

â‡’ 120Â° + mâˆ C = 180Â°

â‡’ mâˆ C = 180Â° - 120Â°

â‡’ mâˆ C = 60Â°

**To construct: **Î”ABC where mâˆ A = 70Â°, mâˆ C = 60Â° and AC = 3 cm.

**Steps of construction:**

(a) Draw a line segment AC = 3 cm.

(b) At point C, draw âˆ YCA = 60Â°.

(c) At point A, draw âˆ XAC = 70Â°

(d) Rays XA and YC intersect at point B

It is the required triangle ABC.

**Answer 4: **

In Î”LMN , mâˆ L - 60Â° mâˆ N = 120Â°, LM = 5 cm

This Î”LMN is not possible to construct because mâˆ L + mâˆ N = 60Â° + 120Â° = 180Â° which forms a linear pair.

**Answer 5: **

Î” ABC, BC = 2 cm, AB = 4 cm and AC = 2 cm

This Î” ABC is not possible to construct because the condition is

Sum of lengths of two sides of a triangle should be greater than the third side.

AB< BC + AC

â‡’ 4 < 2 + 2

â‡’ 4 = 4,

**Answer 6: **

**To construct:** Î”PQR where PQ = 3.5 cm, QR = 4 cm and PR = 3.5 cm

**Steps of construction:**

(a) Draw a line segment QR = 4 cm.

(b) Taking Q as centre and radius 3.5 cm, draw an arc.

(c) Similarly, taking R as centre and radius 3.5 cm, draw another arc which intersects the first arc at point P.

it is the required triangle QPR.

**Answer 7: **

**To construct: **A triangle whose sides are XY = 3 cm, YZ = 4 cm and XZ = 5 cm.

**Steps of construction: **

(a) Draw a line segment ZY = 4 cm.

(b) Taking Z as center and radius 5 cm, draw an arc.

(c) Taking Y as centre and radius 3 cm, draw another arc.

(d) Both arcs intersect at point X.

It is the required triangle XYZ.

**Answer 8: **

**To construct:** A triangle DEF whose sides are DE = 4.5 cm, EF = 5.5 cm and DF = 4 cm.

**Steps of construction: **

(a) Draw a line segment EF = 5.5 cm.

(b) Taking E as center and radius 4.5 cm, draw an arc.

(c) Taking F as center and radius 4 cm, draw another arc which intersects the first arc at point D.

It is the required triangle DEF.

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