Question 7 - Figure it out - Page 157-159 - Chapter 7 Class 8 - Area (Ganita Prakash II)

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Question 7 If M and N are the midpoints of XY and XZ, what fraction of the area of ΔXYZ is the area of ΔXMN? [Hint: Join NY]This problem uses a fundamental rule of geometry: If two triangles have equal bases and share the same height, they have the exact same area. Let's use the hint and follow the steps: For ∆ XNY & ∆ ZNY Let’s draw Height from vertex Y Both triangles have same Height YD And, Base of ∆XNY is XN Base of ∆ZNY is ZN As N is mid-point of XZ, ∴ XN = ZN Since both ∆ XNY & ∆ ZNY have same height and base Their areas are equal So, we can write Area ∆ XNY = Area ∆ ZNY And total Area is ∆ XYZ, so these are half of Area ∆ XYZ So, Area ∆ XNY = Area ∆ ZNY = 𝟏/𝟐 × Area ∆ XYZ For ∆ XMN So, we can write Area ∆ XNY = Area ∆ ZNY And total Area is ∆ XYZ, so these are half of Area ∆ XYZ So, Area ∆ XNY = Area ∆ ZNY = 𝟏/𝟐 × Area ∆ XYZ For ∆ XMN & ∆ YMN Let’s draw Height from vertex N Both triangles have same Height NE And, Base of ∆XMN is XM Base of ∆YMN is YM As M is mid-point of XY, ∴ XM = YM Since both ∆ XMN & ∆ YMN have same height and base Their areas are equal So, we can write Area ∆ XMN = Area ∆ YMN And total Area is ∆ XNY, so these are half of Area ∆ XNY So, Area ∆ XMN = Area ∆ YMN = 𝟏/𝟐 × Area ∆ XNY Let’s find Area ∆ XMN in terms of Area ∆ XYZ Area ∆ XMN = 1/2 × Area ∆ XNY Since Area ∆ XNY = 1/2 × Area ∆ XYZ Area ∆ XMN = 𝟏/𝟐 × 𝟏/𝟐 × Area ∆ XYZ Area ∆ XMN = 𝟏/𝟒 × Area ∆ XYZ So, we can write Area ∆ XMN = Area ∆ YMN And total Area is ∆ XNY, so these are half of Area ∆ XNY So, Area ∆ XMN = Area ∆ YMN = 𝟏/𝟐 × Area ∆ XNY Let’s find Area ∆ XMN in terms of Area ∆ XYZ Area ∆ XMN = 1/2 × Area ∆ XNY Since Area ∆ XNY = 1/2 × Area ∆ XYZ Area ∆ XMN = 𝟏/𝟐 × 𝟏/𝟐 × Area ∆ XYZ Area ∆ XMN = 𝟏/𝟒 × Area ∆ XYZ