Question

Find the length of a transverse common tangent of the two circles whose radii are \(3 \cdot 5 \mathrm{~cm}, 4 \cdot 5 \mathrm{~cm}\) and the distance between their centres is \(10 \mathrm{~cm}\). (1) \(36 \mathrm{~cm}\) (2) \(6 \mathrm{~cm}\) (3) \(64 \mathrm{~cm}\) (4) \(8 \mathrm{~cm}\)

Short answer

Answer: A transverse common tangent between the two circles doesn't exist in this case.

Step-by-step solution

Identify the given values

We are given: - Radius of the first circle: \(3 \cdot 5 \mathrm{~cm}\) - Radius of the second circle: \(4 \cdot 5 \mathrm{~cm}\) - Distance between the centers: \(10 \mathrm{~cm}\)

Calculate the radii of the circles

We need to find the actual radii of the circles by multiplying them by the given constants. \(r_1 = 3 \cdot 5 = 15 \mathrm{~cm}\) \(r_2 = 4 \cdot 5 = 20 \mathrm{~cm}\)

Apply the formula

For the transverse common tangent, the formula is: \(TCT = \sqrt{d^2 - (r_1 + r_2)^2}\) - Replace \(d\), \(r_1\), and \(r_2\) with their respective values: \(TCT = \sqrt{10^2 - (15 + 20)^2}\)

Simplify the equation

Now, we need to simplify and solve the equation: \(TCT = \sqrt{100 - (35)^2}\) \(TCT = \sqrt{100 - 1225}\)

Evaluate the result

The result inside the square root is negative. Since the square root of a negative number is not possible in real numbers, it means that a transverse common tangent between the two circles doesn't exist in this case. So none of the options given in the exercise are correct.

Key concepts

Circle Geometry

Circle geometry involves a variety of elements such as points, lines, and angles associated with a circle. In relation to the transverse common tangent, circle geometry requires understanding of not just one, but the relationship between two circles. The tangent itself is a line that touches a circle at exactly one point, and in the case of two circles, a transverse tangent will touch each circle once, at different points.

In terms of circle geometry, the tangent line’s interaction dictates a fixed relationship with the circle's radius - that is, the radius of the circle at the point of tangency is perpendicular to the tangent. This perpendicular relationship is critical for deducing further properties and is often the key to solving many circle geometry problems.

Radius and Tangent

The radius of a circle is a line segment from the center to any point on the circle. Every point on a circle is equidistant from the center, and this distance is the radius. In relation to tangents, the fundamental property to remember is that at the point where a tangent touches the circle, it is perpendicular to the radius.

When there are two circles with different radii, and we're interested in finding the length of a tangent that touches both circles (known as the transverse common tangent), understanding the individual radius lengths, along with the distance between the two centers, is paramount. In our problem, we must multiply the given constants with the coefficients to find the radii of the two circles—15 cm and 20 cm, respectively—before attempting to apply the tangent length formula.

Applying Mathematical Formulas

Applying mathematical formulas correctly is essential in solving geometry problems. For a problem involving finding the length of a transverse common tangent, we use a specific formula: \( TCT = \sqrt{d^2 - (r_1 + r_2)^2} \), where \( d \) is the distance between the centers of the two circles, and \( r_1 \) and \( r_2 \) are the radii of the respective circles. This formula derives from the Pythagorean theorem, which relates the sides of a right triangle.

It's important to recognize whether the values inputted into the formula will yield a realistic result—in this case, because the result is a square root of a negative number, it's a signal that the scenario is physically impossible; there is no real number length of a tangent that connects two circles with the given parameters. Such critical thinking is vital in mathematical problem-solving and underscores the importance of understanding the implications of formulas, not just their mechanics.