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## Geometry and Mensuration: Test 14

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Question 1 |

The difference between the exterior and interior angles at a vertex of a regular polygon is 150

^{o}. The number of sides of the polygon is10 | |

15 | |

24 | |

30 |

Question 1 Explanation:

Question 2 |

The length of the diagonal BD of the parallelogram ABCD is 18 cm. If P and Q are the centroid of the Î”ABC and Î”ADC respectively then the length of the line segment PQ is

4 cm | |

6 cm | |

9 cm | |

12 cm |

Question 3 |

ABCD is a parallelogram where AB is parallel to CD. BC is produced to Q such that BC= CQ, where P is the point on DC

area Î”( BCP)= areaÎ” ( DPQ) | |

area Î”( BCP) >areaÎ” ( DPQ) | |

areaÂ Î”( BCP) | |

area Î”( BCP) +areaÎ”( DPQ) |

Question 3 Explanation:

We know that: Triangles Î”APC and Î”BCP have the common base PC.

Also, the two triangles Î”APC and Î”BCP are located between the same two parallel lines: AB and PC

Thus, we can say: AD||CQ and AD=CQâ€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦.i

From the above, we can conclude ADQC is a parallelogram.

Also, triangles Î”ADC and Î”DAQ have the common base: AD

Also, the two triangles Î”ADC and Î”DAQ are located between the same two parallel lines: AD and CQ.

Thus, we can conclude: Area of Î”( ADC)= Area of Î”( ADQ)

Letâ€™s subtract area of ( DADP) from both sides: area of Î”( APC)= area of Î”( DPQ)â€¦â€¦..(ii)

From (i) and (ii), we can conclude: area of Î”( BPC) = area of Î”( DPQ)

Question 4 |

ABCD is a rhombus. A straight line through C cuts AD produced at P and AB produced at Q. if DP= Â˝ AB, Then the ratio of the lengths of BQ and AB is

2: 1 | |

1: 2 | |

1: 1 | |

3: 1 |

Question 5 |

In a quadrilateral ABCD, with unequal sides if the diagonals AC and BD intersect at right angles, then

AB ^{2} +BC^{2}= DC^{2} +DA^{2} | |

AB ^{2} +CD^{2} = BC^{2}+ DA^{2} | |

AB ^{2} +AD^{2}= BC^{2}+ CD^{2} | |

AB ^{2}+ BC^{2} =2(CD^{2} +DA^{2}) |

Question 5 Explanation:

$ \begin{array}{l}Let\text{ }the\text{ }diagonals\text{ }intersect\text{ }at\text{ }O.\\A{{O}^{2}}+B{{O}^{2}}=A{{B}^{2}}~,~~\\B{{O}^{2}}+C{{O}^{2}}=B{{C}^{2}}~\\C{{D}^{2}}=O{{C}^{2}}+O{{D}^{2}}\\A{{D}^{2}}=A{{O}^{2}}+A{{D}^{2}}\\From\text{ }the\text{ }equations\text{ }we\text{ }can\text{ }check\text{ }the\text{ }options\text{ }.\\A{{B}^{2}}+C{{D}^{2}}=B{{C}^{2}}+D{{A}^{2}}=B{{O}^{2}}+O{{D}^{2}}+A{{O}^{2}}+C{{O}^{2}}\\We\text{ }can\text{ }conclude\text{ }that\text{ }correct\text{ }option\text{ }is\text{ }\left( b \right)\end{array}$

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