Question

In a circle \(O\) is the centre and \(\angle C O D\) is right angle. \(A C=B D\) and \(C D\) is the tangent at \(P\). What is the value of \(A C+C P\), if the radius of the circle is 1 metre? (a) \(105 \mathrm{~cm}\) (b) \(141.4 \mathrm{~cm}\) (c) \(138.6 \mathrm{~cm}\) (d) can't be determined

Short answer

Answer: (d) can't be determined

Step-by-step solution

Create a diagram and add relevant information

Draw a circle with center O, and create a tangent line at point P. Label points C, D, and A such that AC = BD and angle COD is a right angle. Finally, since we are given that the radius is 1 metre, label the radius OP as 1.

Find the length of OC

We know that angle COD is a right angle, which means triangle COD is a right triangle. OC is the hypotenuse of this triangle. Using the Pythagorean theorem, we have: \[OC^2 = CD^2+OD^2\] Since CD is tangent to the circle at P, and OP is the radius, by the properties of circles, we know that the angle CPD is right angle. Thus, triangle CPD is also a right triangle, so the length of CD is equal to the length of the radius OP = 1. OD is also equal to the length of the radius, so OD = 1 metre. Now we can plug in the known lengths of the sides: \[OC^2 = 1^2+1^2 = 1+1 = 2\] Taking square root we get, OC = √2 metres.

Use properties of triangles to find AC

Since AC = BD and OD = OC, triangle ADC is isosceles with AC = CD. Therefore, AC = CD = 1 metre.

Find the length of PC

Since CP is perpendicular to the tangent line CD at P and OD is the radius, triangle CPD is a right triangle. Using the Pythagorean theorem, we have: \[PC^2 = CP^2 = PD^2 - CD^2\] We know that PD = OC and CD = 1 metre. Thus, \[PC^2 = OC^2 - CD^2 = 2 - 1 = 1\] Taking square root, we get, PC = √1 = 1 metre.

Find the sum of AC and CP

Now, we can find the sum of the lengths of AC and CP: \[AC + CP = 1 + 1 = 2\] Thus, the sum of the lengths of AC and CP is 2 metres, which is equal to 200 cm. The correct answer is not listed among the given choices, so the answer is (d) can't be determined.

Key concepts

Tangent to a Circle

Understanding a tangent's relationship to a circle is fundamental in geometry. A tangent to a circle is a straight line that touches the circle at exactly one point—this point is known as the point of tangency. For any point P that lies on the circle, if you draw a line that just 'kisses' the circle at P, this line is a tangent.

An important property to remember is that a tangent to a circle is perpendicular to the radius at the point of tangency. This means that if you have a circle with center O and a tangent line touching the circle at point P, then the line OP (the radius) is at a right angle to the tangent line at P. In relation to our original exercise, CD is the tangent at point P, and thus triangle CPD forms a right angle at P. This insight is pivotal in solving circle geometry problems as it allows the use of the Pythagorean theorem to find unknown distances related to the circle and the tangent line.

Pythagorean Theorem

The Pythagorean theorem is a staple of geometry, providing a relationship between the lengths of the sides of a right triangle. According to it, in a right-angled 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.

The theorem can be expressed as the formula: \[a^2 + b^2 = c^2\] where c represents the length of the hypotenuse, and a and b represent the lengths of the triangle's other two sides. This theorem is especially useful for solving various geometric problems, such as finding the distance between two points or the length of a line segment. In the exercise at hand, the Pythagorean theorem is used twice: first in step 2 to find the length of OC, and then in step 4 to find the length of CP.

Isosceles Triangle Properties

An isosceles triangle is characterized by having at least two sides that are equal in length, known as the legs. The angles opposite these equal sides are also equal to each other. This symmetry makes the isosceles triangle a helpful tool in solving many geometric problems.

Moreover, if an isosceles triangle has a third side (the base) that is different from the legs, the angle bisector, the median, and the altitude that descend from the angle opposite the base (the vertex angle) to the base are all the same line. This implies that in an isosceles triangle with a right angle, the altitude from the right angle vertex is also the median and the angle bisector.

In the context of our problem, triangle ADC is isosceles, with AC and CD being the equal sides. Since we know the lengths of these sides, using the properties of isosceles triangles, we deduce that AC equals CD, which equals 1 metre (step 3). This property is equally important to validate that the triangle's form can help us solve for other unknown measures in geometric problems.