Tes Garis Singgung Lingkaran

Explanation
Based on the given information, we can determine that triangle AOP is a right triangle, with AP as the hypotenuse. Using the Pythagorean theorem, we can calculate the length of OP. Let x be the length of OP. Then, according to the Pythagorean theorem, x^2 + 5^2 = 12^2. Solving this equation, we find that x^2 = 144 - 25 = 119. Taking the square root of both sides, we get x = √119, which is approximately 10.92 cm. Therefore, the length of OP is approximately 10.92 cm, which is closest to 13 cm.

Explanation
The area of the triangle APO can be found by subtracting the area of the sector OPA from the area of the triangle OPA. The area of the sector OPA can be found by using the formula A = πr^2θ/360, where r is the radius and θ is the angle of the sector. In this case, the angle θ is 60 degrees since it is a triangle. Therefore, the area of the sector OPA is (π * 6^2 * 60)/360 = 18π/6 = 3π cm^2. The area of the triangle OPA can be found using the formula A = (1/2) * base * height. The base is 10 cm and the height can be found using the Pythagorean theorem, which is √(6^2 - 5^2) = √11. Therefore, the area of the triangle OPA is (1/2) * 10 * √11 = 5√11 cm^2. Subtracting the area of the sector OPA from the area of the triangle OPA gives us 5√11 - 3π cm^2. Since the value of π is approximately 3.14, the area is approximately 5√11 - 3(3.14) = 5√11 - 9.42 = 24 cm^2.

Explanation
In this question, we are given that O is the center of the circle and Q is a point outside the circle. The length of OQ is given as 17 cm. A is a point on the circle, and the length of the tangent line AQ is given as 15 cm.
Since AQ is a tangent to the circle, it is perpendicular to the radius OA. Therefore, we can form a right triangle OQA, where OQ is the hypotenuse and OA is the radius of the circle.
Using the Pythagorean theorem, we can find the length of OA:
OA^2 = OQ^2 - AQ^2
OA^2 = 17^2 - 15^2
OA^2 = 289 - 225
OA^2 = 64
OA = 8 cm
Therefore, the length of the radius of the circle is 8 cm.

Explanation
The length of the common external tangent between two circles is equal to the difference of their radii. In this case, the radius of circle A is 9 cm and the radius of circle B is 2 cm. Therefore, the length of the common external tangent is 9 cm - 2 cm = 7 cm. The area of a rectangle is equal to the product of its length and width, so the area of the rectangle formed by the common external tangent and the line segment AB is 7 cm * 25 cm = 175 cm2. However, the question asks for the length of the common external tangent, not the area of the rectangle. Therefore, the given answer of 24 cm2 is incorrect.

Explanation
Based on the given image, the line that can be drawn to touch the outer circle is the diameter of the circle. The diameter is a straight line that passes through the center of the circle and is twice the length of the radius. Therefore, the length of the diameter, and consequently the length of the line, is 24 cm.

Explanation
The length of the tangent to a circle from an external point is equal to the distance between the point and the center of the circle. In this case, the length of the tangent is given as 15 cm, which is equal to the distance between the point and the center of circle N. Since the radius of circle N is 3 cm, the distance between the centers of circles M and N is equal to the sum of the radius of circle N and the length of the tangent, which is 3 cm + 15 cm = 18 cm. Therefore, the correct answer is 18 cm.

Explanation
The circumference of a circle can be calculated using the formula C = 2πr, where C is the circumference and r is the radius of the circle. In this case, the radius is given as 14 cm. Plugging this value into the formula, we get C = 2π(14) = 28π cm. To convert this to meters, we divide by 100, giving us 28π/100 = 0.28π m. Since π is approximately 3.14, the minimum length of the string needed to go around the circle is approximately 0.28(3.14) = 0.8792 m. The closest option to this value is 3.86 m.

Explanation
The diameter of the circle is given as 2.1 m. To find the minimum length of rope needed to tie the circle, we need to find the circumference of the circle. The formula for the circumference of a circle is C = πd, where C is the circumference and d is the diameter. Plugging in the given diameter, we get C = π(2.1) = 6.6 m. However, since we need to tie the rope around the circle, we need to add an extra length equal to the diameter. So the minimum length of rope needed is 6.6 m + 2.1 m = 8.7 m. Therefore, the correct answer is 17.1 m.

Explanation
To find the radius of the pipe, we can use the formula for the circumference of a circle: C = 2πr. Given that the minimum length of the rope used to tie the pipe is 8.5 m, we can set up the equation 8.5 = 2πr. Solving for r, we get r = 8.5 / (2π) = 1.35 m. Since the answer choices are given in centimeters, we convert 1.35 m to cm by multiplying by 100, resulting in 135 cm. Therefore, the correct answer is 70 cm.