Important for :

1

C

In ΔABC and A' B' C

A' B' || AB

∠B' = ∠B, ∠A' = ∠A

∴ A' B' = |
1 | AB |

2 |

(Ref: Mid point Theorem which states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.)

∴ Area of Δ A'B'C = |
1 | × B' C × A' B' |

2 |

= | 1 | × | 1 | × BC × | 1 | AB |

2 | 2 | 2 |

= | 1 | ( | 1 | × BC × AB) |

4 | 2 |

= | 1 | (Area of ΔABC) |

4 |

Hence, option C is correct.

2

C

In the given ||gm P and Q are the mid points of the sides BC and CD respectively, and AC and BD are the diagonals.

In ΔABC, AP is a median and similarly in ΔADC, AQ is a median.

Now, since median divides the triangle into two equal areas, area of ΔABP = 1/2 area of ABC = 1/2 × 1/2 area of the ||gm ABCD

[Since the diagonal AC divides of the ||gm into two equal areas]

= | 1 |
of area of ||gm ABCD ...(1) |

4 |

Similarly, area of ΔADQ

= | 1 |
of area of ||gm ABCD ...(2) |

4 |

Now, in ΔBCD, if we join the mid-points P and Q, the area of the triangle thus formed (ΔPCQ) will be 1/4 of the area of the ΔBCD as per the mid-point theorem.

∴ Area of ΔPCQ = |
1 | of area of ΔBCD |

4 |

= | 1 | of | 1 | of area of ||gm ABCD |

4 | 2 |

= | 1 | of area of ||gm ABCD |

8 |

= (area ||gm ABCD) – [area ΔABP + area ΔADQ + area ΔPCQ]

= (area ||gm ABCD) –

[( | 1 | + | 1 | + | 1 | ) | of area of ||gm ABCD | ] |

4 | 4 | 8 |

= (area ||gm ABCD) – 5/8 × (area of ||gm ABCD)

= | 3 | area ||gm ABCD |

8 |

= | 3 | (2 × area of ΔABCD) |

8 |

= | 3 | of area ΔABC |

4 |

= | 3 |
× 12 = 9 cm^{2} |

4 |

Hence, option C is correct.

3

B

Radius of the larger circle = R cmRadius of the smaller circle = r cm

and π (R

⇒ R^{2} – r^{2} = |
1056 | = | 1056 × 7 | ||

π | 22 |

⇒ R

⇒ (R – r)(R + r)= 336

⇒ (R + r) × 14 = 336

⇒ (R + r) = | 336 | = 24 cm |

14 |

⇒ 2r = 10 ⇒ r = 5 cm.

Hence, option B is correct.

4

5

A

If the ratio of two sides of the gained rectangle is 9:7, the smaller side must be a multiple of 7. But if we closely look at the given options, none except option A; 77 is completely divisible by 7. Hence, we can immediately pick the option 'A' as our answer.

Circumference of circular shape = π × diameter

= | 22 | × 112 = 352 cm |

7 |

Length of wire = 352 cm

⇒ (l + b) = | 352 | = 176 |

2 |

Given ratio of sides = 9 : 7. So,

Smallest side = | 7 | × 176 = 77 cm. |

16 |

Hence, option A is correct.