The angles of a triangle are in the ratio 4 : 5 : 6(i) Is it an acute,

1.	The angles of a triangle are in the ratio 4 : 5 : 6(i) Is it an acute, right or obtuse triangle?(ii) Is it scalene, isosceles or equilateral?
Answer» (i) Given the angles of a triangle are in the ratio 4 : 5 : 6 Sum of three angles of a triangle = 180°. Let the three angles 4x, 5x and 6x 4x + 5x + 6x = 180° 15x = 180° [∵ Vertically opposite angles are equal] ∴ x = 180°/15 ∴ x = 12° ∴ The angles are 4x ⇒ 4 × 12 = 48° 5x ⇒ 5 × 12 = 60° 6x ⇒ 6 × 12 = 72° ∴ The angle of the triangle are 48°, 60°, 72° ∴ It is an acute angles triangle. (ii) We know that the sides opposite to equal angles are equal. Here all the three angles are different. ∴ The sides also different. ∴ The triangle is a scalene triangle.

The angles of a triangle are in the ratio 4 : 5 : 6(i) Is it an acute, right or obtuse triangle?(ii) Is it scalene, isosceles or equilateral?

Answer»

(i) Given the angles of a triangle are in the ratio 4 : 5 : 6 Sum of three angles of a triangle = 180°.

Let the three angles 4x, 5x and 6x

4x + 5x + 6x = 180°

15x = 180° [∵ Vertically opposite angles are equal]

∴ x = 180°/15

∴ x = 12°

∴ The angles are 4x ⇒ 4 × 12 = 48°

5x ⇒ 5 × 12 = 60°

6x ⇒ 6 × 12 = 72°

∴ The angle of the triangle are 48°, 60°, 72°

∴ It is an acute angles triangle.

(ii) We know that the sides opposite to equal angles are equal.

Here all the three angles are different.

∴ The sides also different.

∴ The triangle is a scalene triangle.