Problem 52 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the equation of the circle circumscribed about the right triangle whose vertices are \((0,0),(8,0)\), and \((0,6)\).

Step-by-Step Solution

Answer

The equation of the circumscribed circle is \((x - 4)^2 + (y - 3)^2 = 25\).

Step 1: Identify the Triangle's Right Angle

The vertices given are \((0,0)\), \((8,0)\), and \((0,6)\). The triangle is a right triangle with the right angle at point \((0,0)\).

Step 2: Find the Hypotenuse

The line between the points \((8,0)\) and \((0,6)\) is the hypotenuse of the triangle. To find its length, use the distance formula: \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Substitute the coordinates: \( \sqrt{(8 - 0)^2 + (6 - 0)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \). Thus, the hypotenuse is 10 units long.

Step 3: Determine the Circumcenter

The circumcenter of a right triangle is the midpoint of the hypotenuse. Calculate the midpoint of the points \((8,0)\) and \((0,6)\) using the midpoint formula: \( \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \). Substitute the coordinates: \( \left( \frac{8 + 0}{2}, \frac{0 + 6}{2} \right) = \left( 4, 3 \right) \). Thus, the circumcenter is \((4, 3)\).

Step 4: Use the Circumradius to Write the Circle Equation

The circumradius, which is also the radius of the circumscribed circle, is half the hypotenuse, so it equals 5 units. Use the standard equation of a circle \((x - h)^2 + (y - k)^2 = r^2\) where \((h, k)\) is the center and \(r\) is the radius. Substitute \(h = 4\), \(k = 3\), and \(r = 5\): \((x - 4)^2 + (y - 3)^2 = 25\).

Step 5: Rewrite and Verify the Equation of the Circle

Write the circle equation as \((x - 4)^2 + (y - 3)^2 = 25\). Verify by checking that the given vertices \((0,0)\), \((8,0)\), and \((0,6)\) satisfy this equation. For \((0,0)\): \((0 - 4)^2 + (0 - 3)^2 = 16 + 9 = 25\). Similarly, check the other points: For \((8,0)\), \((8 - 4)^2 + (0 - 3)^2 = 25\), and for \((0,6)\), \((0 - 4)^2 + (6 - 3)^2 = 25\). All points satisfy the equation.

Key Concepts

Right Triangle

A right triangle is a type of triangle that includes one angle measuring 90 degrees. In such triangles, this right angle is key, as it determines many other properties of the triangle.
The right angle makes it possible to use certain mathematical formulas and theorems, like the Pythagorean theorem, which apply specifically to right triangles.
The right triangle we are considering has its vertices at \((0,0)\), \((8,0)\), and \((0,6)\). Here, the right angle is located at the origin point \((0,0)\).

• This geometric property simplifies many calculations related to the triangle and its associated circle.
• Understanding the placement of the right angle helps you identify the other sides and hypotenuse effectively.

Hypotenuse

The hypotenuse is the longest side of a right triangle, stretching directly opposite the right angle. It's a crucial part of right triangles as it's used in various formulas and calculations.
In the given triangle, the vertices are known, allowing us to determine which side serves as the hypotenuse. Points \((8,0)\) and \((0,6)\) define this side.
Using the distance formula, we calculate its length as \(10\) units, a critical step because this length also leads to determining other properties of circumscribed elements:

• Knowing the hypotenuse helps easily find the circumcenter and circumradius.
• It provides the hypotenuse measurement needed for finding other geometric properties.

Circumcenter

The circumcenter of a triangle is the point where the triangle's perpendicular bisectors intersect. However, when dealing with a right triangle, this special point aligns with the midpoint of the hypotenuse.
For our triangle, determining the circumcenter becomes straightforward due to it being a right triangle. The midpoint formula helps in pinpointing this point, giving us the circumcenter location at \((4, 3)\).
This positional information is pivotal because:

• It provides a crucial component in formulating the equation of the circumscribed circle.
• The circumcenter is also the center of the circle passing through all triangle vertices.

Circumradius

The circumradius represents the radius of the circle that is circumscribed about the triangle. For right triangles, the circumradius simplifies to half the hypotenuse length.
So, for the triangle with a hypotenuse measuring \(10\) units, the circumradius calculates as \(5\) units. This is pivotal for the equation of the circumscribed circle:

• It ensures all triangle vertices reside on the circle circumference.
• The circumradius serves as a consistent link between the triangle's geometry and the larger circle's properties.

The equation of the circumscribed circle is derived using this radius and the circumcenter \((4, 3)\), resulting in \((x - 4)^2 + (y - 3)^2 = 25\). This equation not only defines the circle geometrically but also connects the triangle's vertices mathematically to the circle.