Question

Draw a `DeltaABC` in which AB=5 cm, BC= 6 cm and `angleABC=60^(@)`. Construct a triangle similar to ABC with scale factor `(5)/(7)`. Justify the construction.

Answer 1

Steps of construction
1. Draw a line segment AB=5 cm .
2. From point B, draw `angleABY=60^(@)` on which take BC= 6 cm.
3. Join AC , `DeltaABC` is the required triangle.
4. From A, draw any ray AX downwards making an acute angle.
5. Mark 7 point `B_(1),B_(2),B_(3),B_(4),B_(5),B_(6) " and " B_(7)` on AX,
Such that `AB_(1)=B_(1)B_(2)=B_(2)B_(3)=B_(3)B_(4)=B_(4)B_(5)=B_(5)B_(6)=B_(6)B_(7)`.
6. Join `B_(7)B` and from `B_(5)` draw `B_(5)M||B_(7)B` intersecting AB at M.
From point M draw MN||BC intersecting AC at N.
Then, `DeltaAMN` is the required triangle whose sides are equal to `(5)/(7)` of the corresponding sides of the `DeltaABC`.
Justification
Here , `B_(5)M||B_(7)B` ( by construction )
`:. (AM)/(MB)=(5)/(2)`
Now, `(AB)/(AM)=(AM+MB)/(AM)`
`=1+(MB)/(AM)=1+(2)/(5)=(7)/(5)`
Also, MN||BC
`:. DeltaAMN ~DeltaABC`
Therefore, `(AM)/(AB)=(AN)/(AC)=(NM)/(BC)=(5)/(7)`