Question

ANSWER NEEDED

A triangle has sides measuring 12 cm and 18 cm. The angle opposite the 12-cm side measures 28°. Which statement is TRUE given that all angle measurements are rounded to the nearest degree?

A


There are two possible values for the measure of the largest angle of the triangle, and 107° is the greatest.

B


There is only one possible value for the measure of the largest angle of the triangle, 107°.

C


There are two possible values for the measure of the largest angle of the triangle, and 135° is the greatest.

D


There is only one possible value for the measure of the largest angle of the triangle, 135°.

Answer 1

The sum of all angles in a triangle must equal 180°. Since we are provided with one angle measuring 28°, and without the length of the third side, we cannot determine if there are one or two possible triangles that could be formed, nor can we accurately establish the largest angle. Options suggesting an angle greater than 135° are incorrect.

Step-by-step explanation:

A triangle with sides measuring 12 cm and 18 cm and an angle opposite the 12-cm side measuring 28° has been described. According to triangle properties, the sum of angles in any triangle is 180°. Since we already know one angle is 28°, the sum of the other two angles must be 180° - 28° = 152°. The 18-cm side is longer than the 12-cm side, so it must be opposite a larger angle. However, without knowing the third side, we cannot determine the exact measurement of the other angles. But we can conclusively say there cannot be an angle greater than 135° in a triangle if another angle is already 28° because 135° + 28° exceeds 180°, which is impossible in a triangle.

The fact that option C suggests an angle of 135°, when added to 28°, results in 163°, which is not possible as it would exceed the total 180° available in a triangle. The correct answer must be either option A or B. To solve for the largest angle, we could use the Law of Sines if the third side were known. Since this is not the case, and no further information is given to establish the existence of two different triangles (ambiguity case) with the current data set, we can't definitively choose between options A and B without additional information.

Answer 2

Step-by-step explanation:

The diagram of the triangle is shown in the attached photo. Since the length of two sides and one angle is known, we will apply the sine rule. It states that

a/sin A = b/sin B = c/sin C

From the triangle

b = 12

c = 18

12/sin 28 = 18/sinC

Cross multiplying

12SinC = 18sin28

12SinC = 18×0.4695

12sinC = 8.451

SinC = 8.451/12 = 0.70425

C = Sin^-1(0.70425) = 44.77

Since the sum of the angles in a triangle is 180 degrees,

Angle A = 180 - 28 - 44.77 = 107

Therefore, the following statements are true

There is only one possible value for the measure of the largest angle of the triangle, 107°.