points A,B,C and D lie on a straight line in the given order. AC=25cm and BD=46cm. the ratio of length CD to length AB is 5/2. find the length of line segment BC in cm.
make a sketch on a number line of the given.
let BC = x
then AB = 25-x and CD = 46 -x
CD/AB = 5/2
(46-x)/(25-x) = 5/2
92 - 2x = 125-5x
3x = 33
x = 11
So BC = 11 cm
check:
then AB = 25-11= 14 and CD = 46-11 = 35
ration of CD/AB = 35/14 = 5/2
To find the length of line segment BC, we first need to determine the lengths of line segments AB and CD.
Given that AC = 25 cm and BD = 46 cm, we know that the total length of the line segment is equal to the sum of these two segments, which is AB + BC + CD.
Since we are given that the ratio of CD to AB is 5/2, we can set up the following equation:
CD / AB = 5 / 2
To solve for CD, we can rewrite the equation as:
CD = (5 / 2) * AB
Now, we can substitute this value of CD into the equation for the total length of the line segment:
AB + BC + (5 / 2) * AB = 25 + 46
To simplify, we can combine the like terms:
(2 / 2) * AB + BC + (5 / 2) * AB = 71
Now, we can further simplify:
(7 / 2) * AB + BC = 71
To isolate BC, we can subtract (7 / 2) * AB from both sides:
BC = 71 - (7 / 2) * AB
Next, we need to find the value of AB in order to determine BC.
From the equation:
CD / AB = 5 / 2
We can substitute the given values:
(5 / 2) = CD / AB
Cross-multiplying:
2 * CD = 5 * AB
Dividing both sides by 2:
CD = (5 / 2) * AB
Since we know that CD = 25 cm (from AC = 25 cm), we can substitute this value and solve for AB:
25 = (5 / 2) * AB
We can simplify by multiplying both sides by 2:
50 = 5 * AB
Dividing both sides by 5:
AB = 10
Now, we can substitute this value of AB into the equation for BC:
BC = 71 - (7 / 2) * 10
BC = 71 - 35
BC = 36 cm
Therefore, the length of line segment BC is 36 cm.