State which of the following sets are empty sets:
(i) {Triangles whose perimeter is 10 cm and one side is 6 cm}
(ii) {Triangles whose three angles are 40°, 70°, 100°}
(i) {Triangles whose perimeter is 10 cm and one side is 6 cm}
(ii) {Triangles whose three angles are 40°, 70°, 100°}
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Verified answer
Step-by-step explanation:
(i) The set of triangles whose perimeter is 10 cm and one side is 6 cm is an empty set, as such a triangle cannot exist.
To see why, suppose such a triangle did exist. Let its other two sides be x and y. Then we have:
x + y = 10 - 6 = 4
By the triangle inequality, we also have:
x + y > 6
Combining these two inequalities, we get:
4 > 6
Which is a contradiction. Therefore, no such triangle can exist, and the set is empty.
(ii) The set of triangles whose three angles are 40°, 70°, and 100° is not empty, as such a triangle does exist. To see why, note that the sum of the angles in any triangle is always 180°. Therefore, if we add up the three given angles, we get:
40° + 70° + 100° = 210°
Which is equal to the sum of the angles in a triangle. Therefore, a triangle with these angles does exist, and the set is not empty.