Question

In the given figure of circle \(A B\) is the diameter with length \(20 \mathrm{~cm}\) and \(B C\) is \(16 \mathrm{~cm}\), then find the length of \(\mathrm{CO}\) when \(\mathrm{CO}\) is perpendicular on \(A B\) : (a) \(9.6 \mathrm{~cm}\) (b) \(8.4 \mathrm{~cm}\) (c) \(10 \mathrm{~cm}\) (d) data insufficient

Short answer

Answer: Data insufficient.

Step-by-step solution

Identifying the triangle

In the given circle, we can identify a triangle (ΔABC), with AO as one of the legs, OC as the other leg, and hypotenuse AC.

Finding the radius

Since AB is the diameter of the circle, we can divide its length by 2 to find the radius. Radius (AO) = Diameter (AB) / 2 = 20 cm / 2 = 10 cm

Finding length of BO

Now, we know the length of AB is 20 cm and BC is 16 cm. So, the remaining portion of AB which is BO can be calculated as follows: BO = AB - BC = 20 cm - 16 cm = 4 cm

Applying the Pythagorean theorem

Since ΔABC is a right-angled triangle with CO being perpendicular on AB, we can use the Pythagorean theorem to find OC: \(AO^2 = OC^2 + BO^2\) Plug in the known values: \(10^2\mathrm{~cm}^2 = OC^2 + 4^2\mathrm{~cm}^2\) Solve for OC: \(OC^2 = 100\mathrm{~cm}^2 - 16\mathrm{~cm}^2\) \(OC^2 = 84\mathrm{~cm}^2\) \(OC = \sqrt{84\mathrm{~cm}^2}= (\sqrt{84})\mathrm{~cm}\) Since we are not given an option for the non-simplified root answer, we can simplify the radical as follows: \(OC = \sqrt{4*21} = 2\sqrt{21}\) The given options don't include this simplified value, so the answer is (data insufficient).

Key concepts

Understanding Geometry

Geometry is a branch of mathematics that studies the sizes, shapes, properties, and dimensions of objects and spaces. It's fundamental to many fields, from architecture to astronomy. At the heart of geometry is the concept of the point, line, plane, and solid figures like circles, triangles, and squares. Every geometric shape follows certain rules and formulas, making geometry a powerful tool for solving a variety of practical problems.

For instance, in the exercise, we examine geometrical shapes: a circle and a right-angled triangle within it. By understanding how these shapes interact, we can use geometry to solve for unknown lengths and distances, such as the length of line segment CO.

Circle Properties

A circle is a round plane figure whose boundary consists of points equidistant from a fixed center. Some essential properties of a circle that are useful in solving geometric problems include:

• The longest chord of a circle is the diameter, which passes through the center.
• A radius is a line segment from the center to any point on the circle. All radii of a circle have equal length.
• When a diameter is perpendicular to a chord, it bisects the chord and its arc, creating right angles.

In our exercise, AB is the diameter of the circle, and CO is a radius that is perpendicular to the diameter at B, thus creating a right-angled triangle ABC.

Right-Angled Triangles

Right-angled triangles are the cornerstone of trigonometry. They have one angle of exactly 90 degrees and two other, smaller, acute angles. The longest side opposite the right angle is known as the hypotenuse, while the other two sides are called the legs of the triangle.

In our exercise, we focus on triangle ABC where OC is perpendicular to AB, indicating that ABC is a right-angled triangle with OC as one leg, BO as the other, and triangle ABC's hypotenuse as AC. Recognizing these components is crucial to determine unknown lengths within the triangle.

Pythagorean Theorem Application

The Pythagorean theorem is a fundamental principle in geometry, particularly for right-angled triangles. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as:
\[ a^2 + b^2 = c^2 \]
where 'c' represents the length of the hypotenuse, and 'a' and 'b' represent the lengths of the other two sides.

In the context of our exercise, we apply the Pythagorean theorem to find the length of OC (leg of the triangle), with AO (the radius, also a leg) and BO (the remaining part of the diameter after subtracting BC). By applying the theorem, we proved that the given data was insufficient as the answer didn't match any of the provided options.