Explanation:
To find the ratio AB:BC:CD, we can use the information provided about the ratios AB:BD and AC:CD.
Given:
AB:BD = 5:11
AC:CD = 3:1
We can see that the ratio of AB:BD is 5:11, which means that AB is 5 parts of the total, and BD is 11 parts. Similarly, the ratio of AC:CD is 3:1, which means that AC is 3 parts of the total, and CD is 1 part.
Now, we want to find the ratio AB:BC:CD. To do this, we need to express BC in terms of parts of the total.
We know that AB:BD = 5:11, so we can say that AB is 5x parts and BD is 11x parts for some value x.
Similarly, AC:CD = 3:1, so AC is 3y parts, and CD is 1y part for some value y.
Now, let's look at the points in order:
AB:BD = 5:11
AC:CD = 3:1
Now, let's express the parts of the total for each segment:
AB = 5x
BD = 11x
AC = 3y
CD = y
We want to find the ratio AB:BC:CD. BC is the segment between B and C.
AB + BC = BD
We can substitute the expressions we found:
5x + BC = 11x
Now, solve for BC:
BC = 11x - 5x
BC = 6x
So, BC is 6x parts of the total.
Now, let's express the ratio AB:BC:CD:
AB:BC:CD = 5x : 6x : y
We know that AC:CD = 3:1, which means that AC is 3y parts and CD is 1y part. Therefore, y = AC + CD = 3y + y = 4y.
Now, we can express the ratio in terms of y:
AB:BC:CD = 5x : 6x : 4y
To find the ratio in its simplest form, you can divide all parts by the greatest common factor. In this case, the greatest common factor is 1, so the simplest ratio is:
AB:BC:CD = 5x : 6x : 4y
So, the ratio AB:BC:CD is 5x : 6x : 4y.