Find The Circumradius Of A Right Triangle: A Step-by-Step Guide

Hey guys! Today, we're diving into a classic geometry problem: finding the radius of the circle that goes around a right triangle. Specifically, we're given a triangle $ABC$ where $AC = 20$, $BC = 21$, and angle $C$ is a right angle ($90^{\circ}$). Our mission, should we choose to accept it, is to find the radius of the circle that perfectly touches all three vertices of this triangle – the circumcircle. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into calculations, let's break down what we're dealing with. We have a right triangle, which is a triangle containing one angle that measures 90 degrees. The side opposite the right angle is called the hypotenuse, and in our case, it's the side $AB$. The circumcircle is the circle that passes through all the vertices of the triangle. For any triangle, there's only one unique circumcircle. The radius of this circle is what we're trying to find. Knowing that the angle $C$ is 90 degrees is a HUGE hint. It tells us we can use some cool properties specific to right triangles and their circumcircles.

Why Right Triangles are Special

Right triangles have a neat relationship with their circumcircles. The hypotenuse of a right triangle is always a diameter of its circumcircle. This is a fundamental theorem in geometry and simplifies things immensely. So, once we find the length of the hypotenuse $AB$, we can simply divide it by 2 to get the radius of the circumcircle. This makes our task much easier than dealing with general triangles.

Visualizing the Scenario

Imagine drawing the triangle $ABC$ on a piece of paper, with the right angle at vertex $C$. Now, picture a circle drawn around this triangle so that points $A$, $B$, and $C$ all lie on the circle. The center of this circle will lie exactly in the middle of the hypotenuse $AB$. The distance from this center to any of the vertices ($A$, $B$, or $C$) is the radius we're looking for. This visual representation can help solidify the concept and make the problem more intuitive.

Finding the Hypotenuse

Now that we understand the problem, let's calculate the length of the hypotenuse $AB$. Since triangle $ABC$ is a right triangle, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In our case, this translates to:

$AB^2 = AC^2 + BC^2$

We know that $AC = 20$ and $BC = 21$. Plugging these values into the equation, we get:

$AB^2 = 20^2 + 21^2$ $AB^2 = 400 + 441$ $AB^2 = 841$

To find the length of $AB$, we need to take the square root of both sides of the equation:

$AB = \sqrt{841}$ $AB = 29$

So, the length of the hypotenuse $AB$ is 29. Remember, this is the diameter of our circumcircle!

Step-by-Step Calculation

1. Identify the Right Triangle: We're given that triangle $ABC$ has a right angle at $C$.
2. Apply the Pythagorean Theorem: $AB^2 = AC^2 + BC^2$
3. Substitute the Given Values: $AB^2 = 20^2 + 21^2 = 400 + 441 = 841$
4. Solve for the Hypotenuse: $AB = \sqrt{841} = 29$

Calculating the Radius

We know that the hypotenuse $AB$ is the diameter of the circumcircle. Therefore, the radius $R$ of the circumcircle is half the length of the hypotenuse. This can be expressed as:

$R = \frac{AB}{2}$

Since we found that $AB = 29$, we can plug this value into the equation:

$R = \frac{29}{2}$ $R = 14.5$

Therefore, the radius of the circle circumscribed about triangle $ABC$ is 14.5.

Final Answer

The radius of the circle circumscribed about the triangle $ABC$ is 14.5. This result highlights the elegant connection between right triangles and their circumcircles. The simplicity of the calculation—once we recognize the hypotenuse as the diameter—makes this a satisfying problem to solve.

Alternative Approaches and Considerations

While using the property of the right triangle's hypotenuse being the diameter is the most straightforward approach, let's briefly consider other methods. You could, in theory, use the general formula for the circumradius of any triangle, which involves the area of the triangle and the lengths of its sides. However, this would be significantly more complicated and time-consuming for a right triangle.

General Circumradius Formula

The general formula for the circumradius $R$ of a triangle with sides $a$, $b$, and $c$, and area $K$, is:

$R = \frac{abc}{4K}$

In our case, $a = 20$, $b = 21$, and $c = 29$. The area $K$ of a right triangle is simply half the product of its legs:

$K = \frac{1}{2} * AC * BC = \frac{1}{2} * 20 * 21 = 210$

Plugging these values into the general formula, we get:

$R = \frac{20 * 21 * 29}{4 * 210} = \frac{12180}{840} = 14.5$

As you can see, we arrive at the same answer, but the process is much more involved. This reinforces the idea that recognizing the specific properties of right triangles can lead to more efficient solutions.

Key Takeaways

• Right Triangles and Circumcircles: The hypotenuse of a right triangle is the diameter of its circumcircle.
• Pythagorean Theorem: $a^2 + b^2 = c^2$ for a right triangle with legs $a$ and $b$, and hypotenuse $c$.
• Efficiency: Recognizing special properties of geometric figures can significantly simplify problem-solving.

Practice Problems

To solidify your understanding, try these practice problems:

1. Triangle $DEF$ is a right triangle with $DE = 12$, $EF = 5$, and angle $E = 90^{\circ}$. Find the radius of the circumcircle.
2. In right triangle $PQR$, $PQ = 8$, $QR = 15$, and angle $Q = 90^{\circ}$. Determine the circumradius.
3. A right triangle has legs of length 7 and 24. What is the radius of its circumscribed circle?

Conclusion

Finding the circumradius of a right triangle is a classic problem that beautifully illustrates the power of geometric relationships. By understanding the connection between right triangles and their circumcircles, and by applying the Pythagorean theorem, we can solve these problems efficiently and elegantly. So, keep practicing, keep exploring, and keep those geometry skills sharp! You've got this!

Remember, the key to mastering geometry is practice and a solid understanding of fundamental theorems. Keep exploring different types of problems, and don't be afraid to draw diagrams to visualize the situations. Happy solving, and I'll catch you in the next math adventure!