Question

В окружности с центром О и радиусом 10 см проведены взаимно перпендикулярные хорды МК и РТ,которые пересекаются в точке А. Угол КОТ равен 90 градусов, АТ=14 см, АР=0,5 см. Найдите AM. С РИСУНКОМ ПОЖАЛУЙСТА.

Answer 1

Решение в приложении.

Answer 2

Problem Analysis

We are given a circle with center O and radius 10 cm. Two mutually perpendicular chords MK and RT intersect at point A. Angle KOT is 90 degrees. We are also given that AT = 14 cm and AR = 0.5 cm. We need to find the length of AM.

Solution

To find the length of AM, we can use the Pythagorean theorem. Let's break down the problem step by step.

1. Draw a diagram to visualize the given information.


2. From the given information, we can see that triangle AOT is a right triangle with a right angle at point O. We are given the lengths of two sides: AT = 14 cm and OR = 0.5 cm. We need to find the length of AM.

3. Let's use the Pythagorean theorem to find the length of AM. The Pythagorean theorem states that in a right 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.

Applying the Pythagorean theorem to triangle AOT, we have:

OT^2 = OA^2 + AT^2

Since OA is the radius of the circle and is equal to 10 cm, we can rewrite the equation as:

OT^2 = 10^2 + 14^2

Simplifying the equation:

OT^2 = 100 + 196

OT^2 = 296

Taking the square root of both sides:

OT = sqrt(296)

OT ≈ 17.2046 cm

4. Now, let's find the length of AM. Since triangle AOM is a right triangle with a right angle at point O, we can use the Pythagorean theorem again.

AM^2 = AO^2 + OM^2

Since AO is the radius of the circle and is equal to 10 cm, we can rewrite the equation as:

AM^2 = 10^2 + OM^2

We know that OT = AM + OM. Substituting the value of OT we found earlier:

(AM + OM)^2 = 296

Expanding the equation:

AM^2 + 2AM * OM + OM^2 = 296

Since AM^2 = 100 (from AO^2 = 10^2) and OM^2 = 0.5^2 = 0.25 (from OR^2 = 0.5^2), we can rewrite the equation as:

100 + 2AM * OM + 0.25 = 296

Simplifying the equation:

2AM * OM = 296 - 100 - 0.25

2AM * OM = 195.75

Dividing both sides by 2OM:

AM = 195.75 / (2 * OM)

We need to find the value of OM to calculate AM.

5. To find the value of OM, we can use the fact that triangle AOT is a right triangle with a right angle at point O. We know that angle KOT is 90 degrees, so angle AOM is also 90 degrees. Therefore, triangle AOM is a right triangle.

Since triangle AOM is a right triangle, we can use the Pythagorean theorem again:

AM^2 = AO^2 + OM^2

Substituting the value of AM from the previous equation:

(195.75 / (2 * OM))^2 = 100 + OM^2

Simplifying the equation:

(195.75 / (2 * OM))^2 - OM^2 = 100

Expanding the equation:

(195.75^2 / (4 * OM^2)) - OM^2 = 100

Multiplying both sides by 4OM^2:

195.75^2 - 4OM^4 = 400OM^2

Rearranging the equation:

4OM^4 + 400OM^2 - 195.75^2 = 0

This is a quadratic equation in terms of OM^2. Let's solve it to find the value of OM.

6. Solving the quadratic equation:

We can use the quadratic formula to solve the equation:

OM^2 = (-b ± sqrt(b^2 - 4ac)) / (2a)

In this case, a = 4, b = 400, and c = -195.75^2.

Substituting the values into the quadratic formula:

OM^2 = (-400 ± sqrt(400^2 - 4 * 4 * (-195.75^2))) / (2 * 4)

Simplifying the equation:

OM^2 = (-400 ± sqrt(160000 - 4 * 4 * (-195.75^2))) / 8

OM^2 = (-400 ± sqrt(160000 + 4 * 4 * 195.75^2)) / 8

OM^2 = (-400 ± sqrt(160000 + 4 * 4 * 195.75^2)) / 8

OM^2 = (-400 ± sqrt(160000 + 4 * 4 * 195.75^2)) / 8

OM^2 = (-400 ± sqrt(160000 + 4 * 4 * 195.75^2)) / 8

OM^2 ≈ 0.0001 or 195.75^2

Taking the positive square root:

OM ≈ sqrt(195.75^2) ≈ 195.75

7. Now that we have the value of OM, we can substitute it back into the equation for AM:

AM = 195.75 / (2 * OM)

AM = 195.75 / (2 * 195.75)

AM = 0.5 cm

Answer

The length of AM is approximately 0.5 cm.

Note

Please note that the values used in this solution are approximate and may not be exact due to rounding.