Question

Через точку М, находящуюся на расстоянии 15см от центра окружности радиусом 17 см, проведена хорда CD, которая делится точкой М на отрезки CM:MD=1:4. Найдите хорду CD

Answer 1

Хорда делится на отрезки (х и 4х)
диаметр делится на отрезки (2 и 32)
х*4х=2*32
4х"=64
х"=16
х=4
отрезки равны
4 и 16
длина хорды 16+4=20см

Answer 2

Problem Analysis

We are given a circle with a radius of 17 cm and a point M located 15 cm away from the center of the circle. A chord CD is drawn through point M, dividing it into segments CM and MD in a ratio of 1:4. We need to find the length of chord CD.

Solution

To find the length of chord CD, we can use the properties of chords and their segments in a circle.

Let's denote the length of CM as x. Since MD is four times the length of CM, we can express MD as 4x.

According to the properties of chords and their segments, the product of the lengths of the segments of a chord is equal. Therefore, we have:

CM * MD = CD^2

Substituting the values of CM and MD, we get:

x * 4x = CD^2

Simplifying the equation, we have:

4x^2 = CD^2

Taking the square root of both sides, we get:

2x = CD

Therefore, the length of chord CD is equal to twice the length of CM.

To find the length of CM, we can use the Pythagorean theorem. In the right triangle CMO, where O is the center of the circle, we have:

CM^2 + MO^2 = CO^2

Substituting the values, we get:

x^2 + 15^2 = 17^2

Simplifying the equation, we have:

x^2 + 225 = 289

x^2 = 289 - 225

x^2 = 64

Taking the square root of both sides, we get:

x = 8

Therefore, the length of CM is 8 cm.

Finally, substituting the value of CM into the equation for CD, we get:

CD = 2 * CM = 2 * 8 = 16 cm

Answer

The length of chord CD is 16 cm.

Explanation

We can solve this problem by using the properties of chords and their segments in a circle. The length of chord CD is equal to twice the length of segment CM, which is 8 cm. Therefore, the length of chord CD is 16 cm.