Determine whether you can construct many, one, or no triangle(s) with each description.
a)An isosceles triangle with angle measures of 40° and 100°
b)A right triangle with three congruent sides
c)A scalene triangle with side lengths of 7 cm, 3 cm, and 9 cm
So Can you form one triangle or two triangles for number a) and b)
(a) yes, since 40+40+100 = 180
(b) no, since c^2 = a^2+b^2
(c) yes. Use 9 for the base. The other two sides sum to 10, so to make them fit, you have to lift up the vertex to form a triangle.
a) In order to construct a triangle, the sum of any two angles must be greater than the measure of the third angle.
For the given description, an isosceles triangle with angle measures of 40° and 100°, we can find the third angle by subtracting the sum of the given angles from 180°.
Third angle = 180° - (40° + 100°) = 180° - 140° = 40°
Since the third angle is also 40°, the given description can construct only one triangle.
b) A right triangle with three congruent sides is not possible. In a right triangle, one angle measures 90°, while the other two angles are acute (less than 90°). Having three congruent sides would mean that all three angles are equal, which contradicts the definition of a right triangle. Therefore, for this description, it is not possible to construct any triangle.
c) In order to construct a triangle, the sum of any two sides must be greater than the length of the third side.
For the given description, a scalene triangle with side lengths of 7 cm, 3 cm, and 9 cm, let's check the triangle inequality:
7cm + 3cm > 9cm (10cm > 9cm) - True
3cm + 9cm > 7cm (12cm > 7cm) - True
9cm + 7cm > 3cm (16cm > 3cm) - True
Since all three inequalities are true, it is possible to construct a triangle with the given side lengths. Hence, the given description can construct one triangle.
To determine whether you can construct many, one, or no triangles with the given descriptions, you need to apply the triangle inequality theorem. According to this theorem, for a triangle with side lengths a, b, and c:
- The sum of the lengths of any two sides must be greater than the length of the third side.
Let's analyze each case separately:
a) An isosceles triangle with angle measures of 40° and 100°:
In an isosceles triangle, at least two sides have the same length. The sum of the angles in a triangle is always 180°. Since the angles are 40°, 40°, and 100°, the two equal angles must correspond to the equal sides. Thus, the side opposite the 100° angle must be longer than the other two sides combined: 40° + 40° = 80°. Therefore, it is not possible to construct a triangle with these measurements.
b) A right triangle with three congruent sides:
In a right triangle, one of the angles is 90°. If all three sides are congruent, it means that all angles must be 60° (since the sum of angles in a triangle is 180°). However, a triangle with three congruent sides and all angles equal to 60° does not form a right triangle. Therefore, it is not possible to construct a triangle with these measurements.
c) A scalene triangle with side lengths of 7 cm, 3 cm, and 9 cm:
A scalene triangle has three sides of different lengths. To determine if it can be constructed, we need to verify if the sum of the lengths of any two sides is greater than the length of the third side.
- Side 1 + Side 2 > Side 3:
7 cm + 3 cm = 10 cm > 9 cm (True)
- Side 1 + Side 3 > Side 2:
7 cm + 9 cm = 16 cm > 3 cm (True)
- Side 2 + Side 3 > Side 1:
3 cm + 9 cm = 12 cm > 7 cm (True)
In this case, all three inequalities hold true, meaning that it is possible to construct a triangle with side lengths of 7 cm, 3 cm, and 9 cm. Therefore, you can construct one triangle with these measurements.