Animated Solution for Mathematics - Trigonometry: Which of the following pieces of data does NOT uniquely determine an acute-angled triangle $ABC$ ($R$ being the radius of the circumcircle)?

Unlocking the Triangle

Imagine you are standing in a vast, empty field, and your task is to construct a perfect, acute-angled triangle $ABC$ inside a circle of radius $R$. You have a set of tools—some measurements—and you need to know if these tools are enough to 'freeze' the triangle in place.

In geometry, to uniquely determine a triangle, you need three independent pieces of information. If you have fewer, the triangle can morph and change; if you have redundant information, you are just repeating yourself.

Our Master Key

Whenever you see a problem involving sides ($a,b,c$), angles ($A,B,C$), and the circumradius $R$, your intuition should immediately jump to the Sine Law:

This is not just an equation; it is the heartbeat of the triangle. It links the linear dimensions to the angular ones.

It tells us that the ratio of any side to the sine of its opposite angle is constant and equal to the diameter of the circumcircle. This is the bridge we will use to test our options.

Testing the Options

Let's look at the first option: $a,\sin A,\sin B$. We have the side $a$ and the sine of its opposite angle $A$.

Using the Sine Law, we can immediately find the circumradius:

Now that $R$ is locked in, we can find side $b$ using $b=2R\sin B$. Since the triangle is acute, $\sin B$ uniquely determines $\angle B$. With $\angle A$ and $\angle B$ known, $\angle C$ is simply $180^{\circ }-(A+B)$. Everything is fixed!

Next, consider the SSS (Side-Side-Side) case: $a,b,c$. This is the classic congruence criterion.

If you know all three sides, the triangle is rigid. You can use the Cosine Law, to find the angles. The triangle is frozen.

Now, look at $a,\sin B,R$. We can find $\sin A=\frac{a}{2R}$.

Again, because the triangle is acute, $\angle A$ is unique. We then find $b=2R\sin B$, and the rest follows. This also works perfectly.

The Self-Fulfilling Prophecy

Finally, we arrive at the culprit: $a,\sin A,R$. Let's plug these into the Sine Law:

This is not a set of instructions to build a triangle; it is a statement that is always true for any triangle. It provides no new information about the other sides or angles.

If you fix side $a$ and the circumradius $R$, you can slide the third vertex $C$ anywhere along the major arc of the circumcircle. The angle $A$ will remain constant, the side $a$ will remain constant, and $R$ will remain constant, but the triangle itself will change shape infinitely.

The Philosophy of Independence

To determine a triangle, you need three independent pieces of data. Redundant data, like in the case of $a,\sin A,R$, leaves the system underdetermined.

It is a beautiful reminder that in mathematics, as in life, it is not just the quantity of information that matters, but the quality and independence of that information. Keep exploring, keep questioning, and let the geometry guide you!