How many distinct triangles can be drawn given ΔABC where a = 10, c = 16, and ∠A = 30°?

Answer 1

In order to determine the number of distinct triangles that can be formed, we must analyze the given information. From the angle measures of the triangle, we know that the largest side, c, is opposite ∠C, and the second largest side, a, is opposite ∠A. Therefore, side b is opposite ∠B. The sum of the angles in a triangle is always 180°, so we can find ∠B using the equation:

∠B = 180° - ∠A - ∠C
= 180° - 30° - 90° (since ∠C, the largest angle, is a right angle)
= 60°

Given that side a has a length of 10 and side c has a length of 16, we can use the law of cosines to find the length of side b:

b² = a² + c² - 2ac*cos(B)
b² = 10² + 16² - 2(10)(16)*cos(60°)
b² = 100 + 256 - 320*cos(60°)
b² = 356 - 320(0.5)
b² = 356 - 160
b² = 196
b = √196
b = 14

Since we've found that side b has a length of 14, we now have all three sides of the triangle: a = 10, b = 14, and c = 16.

Now, we can determine the number of distinct triangles that can be drawn by varying the positions and orientations of the sides while keeping their lengths constant.

If we fix side a and c in position and change the orientation of side b, we can see that the triangle can be "folded" in both directions, forming either side b or its "mirror image" as the base.
However, if we keep side b fixed and change the positions of sides a and c, we can "rotate" the triangle and create different configurations.

Therefore, we can conclude that there is only one distinct triangle that can be drawn.