Question

FOR 21 POINTS

A triangle has an angle measuring 90°, an angle measuring 20°, and a side that is 6 units long. The 6-unit side is in between the 90° and 20° angles.


How many unique triangles can you draw like this?

Answer 1

With angles of 90°, 20°, and 70°, and given the length of one side, only one unique triangle can be constructed because the side lengths are determined by the angles in trigonometry.

Step-by-step explanation:

When describing a triangle, there are a few fundamental properties to remember. A triangle is a three-sided figure lying on a plane, with the sum of its angles adding up to 180 degrees. Therefore, knowing that the triangle in question has a 90° angle and a 20° angle, we can determine the third angle since the sum of the angles must be 180°. Subtracting the known angles from 180° gives us the third angle:

180° - 90° - 20° = 70°.

This means the triangle has angles of 90°, 20°, and 70° with a side that is 6 units long between the 90° and 20° angles. Such a triangle is uniquely determined because the side opposite the right angle (the hypotenuse) is fixed by the angles and the given side, following the rules of trigonometry. Therefore, only one unique triangle can be drawn with the given measurements.

Answer 2

One

Step-by-step explanation:

Since we have a 90° angle and a 20° angle, the last angle must be 70°.

180° - 90° - 20° = 70°

Now we also have the requirement of a 6-unit side between the 90° and 20° angles.

With all of these requirements, we can only draw one triangle like this since we have an angle, an angle, a required third angle, and a side.