If M and N are the midpoints of XY and XZ, what fraction of the area of ∆XYZ is the area of ∆XMN? [Hint: Join NY] Class 8

If M and N are the midpoints of XY and XZ, what fraction of the area of ∆XYZ is the area of ∆XMN? [Hint: Join NY] Class 8

Question 1.

If M and N are the midpoints of XY and XZ, what fraction of the area of ∆XYZ is the area of ∆XMN? [Hint: Join NY] Class 8

Solution:

Join NY and draw NP ⊥ XY and YQ ⊥ XZ.

Now, area of ∆XMN = [latex]\frac{1}{2}[/latex] × M × NP

= [latex]\frac{1}{2}[/latex] × YM × NP

= Area of ∆YNM.

So, area of ∆XMN = [latex]\frac{1}{2}[/latex]area of ∆XNY. .........(1)

Again, area of ∆XNY = [latex]\frac{1}{2}[/latex] × NX × YQ

= [latex]\frac{1}{2}[/latex] × [latex]\frac{ZX}{2}[/latex] × YQ

= [latex]\frac{1}{2}[/latex] area of ∆XYZ. ......(2)

So, from (1) and (2),

Area of ∆XMN = [latex]\frac{1}{2}[/latex] × [latex]\frac{1}{2}[/latex] of area of ∆XYZ

= [latex]\frac{1}{4}[/latex] of area of ∆XYZ

Question 2.

Gopal needs to carry water from the river to his water tank. He starts from his house. What is the shortest path he can take from his house to the river and then to the water tank? Roughly recreate the map in your notebook and trace the shortest path. Class 8

Solution: