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Question
Draw a triangle ABC in which AB = 5 cm, BC = 6 cm and ∠ABC = 60°. Construct a triangle similar to ∆ABC with scale factor `5/7`. Justify the construction.
Solution
Steps of construction:
- Draw a line segment AB = 5 cm.
- From point B, draw ∠ABY = 60° on which take BC = 6 cm.
- Join AC, ∆ABC is the required triangle.
- From A, draw any ray AX downwards making an acute angle.
- Mark 7 points B1, B2, B3, B4, B5, B6 and B7 on AX, such that AB1 = B1B2 = B2B3 = B3B4 = B4B5 = B5B6 = B6B7.
- Join B7B and from B5 draw B5M || B7B intersecting AB at M.
- From point M draw MN || BC intersecting AC at N. Then, ∆AMN is the required triangle whose sides are equal to `5/7` of the corresponding sides of the ∆ABC.
Justification:
Here, B5M || B7B ...(By construction)
∴ `"AM"/"MB" = 5/2`
Now, `"AB"/"AM" = ("AM" + "MB")/"AM"`
= `1 + "MB"/"AM"`
= `1 + 2/5`
= `7/5`
Also, MN || BC
∴ ∆AMN ∼ ∆ABC
Therefore, `"AM"/"AB" = "AN"/"AC" = "NM"/"BC" = 5/7`
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