Four line segments are graphed on a coordinate plane. All four lines meet at point E.Line segment Upper A Upper E begins at left parenthesis 0 comma 0 right parenthesis and ends at left parenthesis 4 comma 5 right parenthesis. Line segment Upper B Upper E begins at left parenthesis 3 comma 0 right parenthesis and ends at left parenthesis 4 comma 5 right parenthesis. Line segment Upper C Upper E begins at left parenthesis 6 comma 0 right parenthesis and ends at left parenthesis 4 comma 5 right parenthesis. Line segment Upper D Upper E begins at left parenthesis 9 comma 0 right parenthesis and ends at left parenthesis 4 comma 5 right parenthesis.
Question
Multiple Choice
Use the graph to answer the question.
What is start fraction upper A upper D over upper A upper B end fraction in simplest form?
(1 point)
Responses
start fraction 10 over 3 end fraction
Image with alt text: start fraction 10 over 3 end fraction
start fraction 1 over 3 end fraction
Image with alt text: start fraction 1 over 3 end fraction
Start Fraction 17 over 5 End Fraction
Image with alt text: Start Fraction 17 over 5 End Fraction
3
To find the value of the fraction start fraction upper A upper D over upper A upper B end fraction, we need to determine the lengths of line segments AD and AB.
From the graph, we can see that line segment AD starts at point (0,0) and ends at point (9,0). Therefore, the length of AD is 9 units.
Line segment AB starts at point (3,0) and ends at point (4,5). To find the length of AB, we can use the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates, we can calculate the length of AB as follows:
Length AB = √((4 - 3)^2 + (5 - 0)^2)
= √(1^2 + 5^2)
= √(1 + 25)
= √26
Therefore, start fraction upper A upper D over upper A upper B end fraction is equal to 9 / √26.
To simplify the fraction, we need to rationalize the denominator by multiplying both the numerator and denominator by √26:
start fraction upper A upper D over upper A upper B end fraction = 9 / √26 * √26 / √26
= 9√26 / 26
So, the simplified value of start fraction upper A upper D over upper A upper B end fraction is 9√26 / 26.
None of the given options match this value.
pick one from above
Based on the given options, the closest value to 9√26 / 26 is:
Start Fraction 17 over 5 End Fraction
To find the answer to this multiple-choice question, you need to compare the lengths of line segments A to D and A to B.
The length of a line segment can be found using the distance formula:
d = √((x2-x1)^2 + (y2-y1)^2)
For line segment AD:
(x1, y1) = (0, 0)
(x2, y2) = (9, 0)
Using the distance formula, we have:
d(AD) = √((9-0)^2 + (0-0)^2)
= √(81 + 0)
= √81
= 9
For line segment AB:
(x1, y1) = (0, 0)
(x2, y2) = (4, 5)
Using the distance formula, we have:
d(AB) = √((4-0)^2 + (5-0)^2)
= √(16 + 25)
= √41
So, the ratio of AD to AB is:
AD/AB = 9/√41
To simplify this fraction, we need to rationalize the denominator. Multiply both the numerator and denominator by √41:
AD/AB = (9/√41) * (√41/√41)
= 9√41/41
The simplified fraction is 9√41/41.
From the given options, the closest answer is not provided. If you had to select an option, the closest one to the answer would be 'None of the above.'