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प्रश्न
Draw a triangle ABC in which AB = 4 cm, BC = 6 cm and AC = 9 cm. Construct a triangle similar to ∆ABC with scale factor `3/2`. Justify the construction. Are the two triangles congruent? Note that all the three angles and two sides of the two triangles are equal.
उत्तर
Steps of construction:
- Draw a line segment BC = 6 cm.
- Taking B and C as centres, draw two arcs of radii 4 cm and 9 cm intersecting each other at A.
- Join BA and CA, ΔABC is the required triangle.
- From B, draw any ray BX downwards making an acute angle.
- Mark three points B1, B2, B3 on BX, such that BB1 = B1B2 = B2B3.
- Join B2C and from B3 draw B3M || B2C intersecting the extended line segment BC at M.
- From point M, draw MN || CA intersecting the extended line segment BA to N.
Then, ΔNBM is the required triangle whose sides are equals to `3/2` of the corresponding sides of the ΔABC.
Justification:
Here, B3M || B2C
∴ `"BC"/"CM" = 2/1`
Now, `"BM"/"BC" = ("BC" + "CM")/"BC"`
= `1 + "CM"/"BC"`
= `1 + 1/2`
= `3/2`
Also, MN || CA
∴ ΔABC ∼ ΔNBM
Therefore, `"NB"/"AB" = "NM"/"AC" = "BM"/"BC" = 3/2`
The two triangles are not congruent because, if two triangles are congruent, then they have same shape and same size. Here, all the three angles are same but three sides are not same i.e., one side is different.
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संबंधित प्रश्न
Construct a triangle ABC in which AB = 5 cm, BC = 6 cm and ∠ABC = 60˚. Now construct another triangle whose sides are 5/7 times the corresponding sides of ΔABC.
Construct a Δ ABC in which AB = 6 cm, ∠A = 30° and ∠B = 60°, Construct another ΔAB’C’ similar to ΔABC with base AB’ = 8 cm.
Draw a right triangle ABC in which AC = AB = 4.5 cm and ∠A = 90°. Draw a triangle similar to ΔABC with its sides equal to (5/4)th of the corresponding sides of ΔABC.
Find the co-ordinates of the points of trisection of the line segment AB with A(2, 7) and B(–4, –8).
Choose the correct alternative:
ΔPQR ~ ΔABC, `"PR"/"AC" = 5/7`, then
Choose the correct alternative:
∆ABC ∼ ∆AQR. `"AB"/"AQ" = 7/5`, then which of the following option is true?
If the point P (6, 7) divides the segment joining A(8, 9) and B(1, 2) in some ratio, find that ratio
Solution:
Point P divides segment AB in the ratio m: n.
A(8, 9) = (x1, y1), B(1, 2 ) = (x2, y2) and P(6, 7) = (x, y)
Using Section formula of internal division,
∴ 7 = `("m"(square) - "n"(9))/("m" + "n")`
∴ 7m + 7n = `square` + 9n
∴ 7m – `square` = 9n – `square`
∴ `square` = 2n
∴ `"m"/"n" = square`
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The basic principle used in dividing a line segment is ______.
Draw an isosceles triangle ABC in which AB = AC = 6 cm and BC = 5 cm. Construct a triangle PQR similar to ∆ABC in which PQ = 8 cm. Also justify the construction.
