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Answer :

Steps of Construction the figure by the given data:

Step 1. Construct a \(\triangle \) ABC, with BC = 4 cm, CA = 3 cm and ∠BCA = 90°.

Step 2. By making an acute angle with BC, draw a ray BX .

Step 3. Mark five points \(B_1, B_2, B_3, B_4 \)and \(B_5 \)on BX, such that \(BB_1 = B_1B_2 = B_2B_3 = B_3B_4 = B_4B_5\) and join \(B_3C\).

Step 6. Through \(B_5\), draw \(B_5C’\) parallel to \(B_3C\) intersecting BC produced at C’.

Step 7. Through C’, draw C’A’ parallel to CA intersecting AB produced at A’.

Thus, \(\triangle \) A’BC’ is the required right triangle.

**JUSTIFICATION:**

In \(ΔA’BC’ and ΔABC\),

\(∠ABC=∠ A’BC’ \)(common)

\(∠ACB=∠ A’C’B\) (corresponding angles)

\(∴ ΔABC∼ΔA’BC’ \) (By AA similarity)

\(∴ \frac{AB}{A’B}=\frac{AC}{A’C’}=\frac{BC}{BC’}\)

But, \(\frac{BC}{BC’}=\frac{BB_3}{BB_5}=\frac{3}5 ∴ \frac{BC’}{BC}=\frac{5}3 \)

\(⇒ \frac{A’B}{AB}=\frac{A’C’}{AC}=\frac{BC’}{BC}=\frac{5}3 .\)

\(\therefore \) Sides of the new triangle formed are \(\frac{5}{3} \) times the corresponding sides of the first triangle.

- Draw a line segment of length 7.6 cm and divide it in the ratio 5:8. Measure the two parts.
- Construct a triangle of sides 4 cm, 5 cm and 6 cm and then a triangle similar to it whose sides are \(\frac{2}{3} \) of the corresponding sides of the first trianngle.
- Construct a triangle with sides 5 cm, 6 cm and 7 cm and then another triangle whose sides are \(\frac{7}{5} \) of the corresponding sides of the first triangle.
- Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then another triangle whose sides are \(1\frac { 1 }{ 2 }\) times the corresponding sides of the isosceles triangle.
- Draw a triangle ABC with side BC = 6 cm, AB = 5 cm and \(\angle \)ABC=60° ;. Then construct a triangle whose sides are \(\frac{3}4\) of the corresponding sides of the triangle ABC.
- Draw a triangle ABC with side BC = 7 cm, \(\angle \) B=45°, \(\angle \) A=105°;. Then construct a triangle whose sides are \(\frac{4}3 \) times the corresponding sides of \(\triangle \) ABC.
- Q.1In each of the following, give the justification of the construction also: 1. Draw a line segment of length 7.6 cm and divide it in the ratio 5 : 8. Measure the two parts.
- Q.2 Construct a triangle of sides 4 cm, 5 cm and 6 cm and then a triangle similar to it whose sides are \(\frac{2}{3} \) of the corresponding sides of the first triangle.
- Q.3 Construct a triangle with sides 5 cm, 6 cm and 7 cm and then another triangle whose sides are \(\frac{7}{5} \) of the corresponding sides of the first triangle
- Q.4 Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then another triangle whose sides are \(1\frac{1}{2} \) times the corresponding sides of the isosceles triangle.
- Q.5 Draw a triangle ABC with side BC = 6 cm, AB = 5 cm and \(\angle \)ABC = 60°. Then construct a triangle whose sides are \(\frac{3}{4} \) of the corresponding sides of the triangle ABC.
- Q.6 Draw a triangle ABC with side BC = 7 cm, \(\angle \) B = 45°, \(\angle \) A = 105°. Then, construct a triangle whose sides are \(\frac{4}{3} \)times the corresponding sides of \(\triangle \)ABC.
- Q.7 Draw a right triangle in which the sides (other than hypotenuse) are of lengths 4 cm and 3 cm. Then construct another triangle whose sides are \(\frac{5}{3} \) times the corresponding sides of the given triangle.

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