Question

Хорды МN и РК пересекаются в точке А так, что МА=3 см, NА=16 см, РА:КА=1:3.Найдите длину хорды РК

Answer 1

МА х НА = РА х КА , 3 х 16 = 1а х 3а

48 = 3а в квадрате

а= 4 = РА

КА = 3 х 4 = 12

РК = 4+12=16

Answer 2

Problem Analysis

We are given that the chords MN and RK intersect at point A, with MA = 3 cm, NA = 16 cm, and the ratio of RA to KA is 1:3. We need to find the length of chord RK.

Solution

To find the length of chord RK, we can use the properties of intersecting chords in a circle. According to the intersecting chord theorem, when two chords intersect in a circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

Let's denote the length of segment RA as x and the length of segment KA as 3x (since the ratio of RA to KA is 1:3). We can set up the following equation:

MA * NA = RA * KA

Substituting the given values, we have:

3 cm * 16 cm = x * 3x

Simplifying the equation:

48 cm^2 = 3x^2

Dividing both sides by 3:

16 cm^2 = x^2

Taking the square root of both sides:

x = 4 cm

Therefore, the length of segment RA is 4 cm and the length of segment KA is 12 cm.

To find the length of chord RK, we can use the Pythagorean theorem. The length of chord RK is equal to the sum of the lengths of segments RA and KA:

RK = RA + KA = 4 cm + 12 cm = 16 cm

Therefore, the length of chord RK is 16 cm.

Answer

The length of chord RK is 16 cm.