Three points A,B and C have coordinates (1,0),(4,4) and (13,5) respectively.
1.By how many units does the length of AC exceed the length of AB
2.What is the slope of BC
1. The length of AC is
sqrt[(13-1)^2 + (5-0)^2]
= sqrt169 = 13
The length of AB is
sqrt[(4-1)^2 + (4-0)^2]
= sqrt(25) = 5
The difference in lengths is 8.
2. BC slope = delty/deltax
= (5-4)/(13-4) = 1/9
To find the answers, we'll use the distance formula and slope formula.
1. Length of AC exceeds the length of AB:
To calculate the distance between two points (x1, y1) and (x2, y2), we use the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Length of AB:
A(1, 0) and B(4, 4)
Distance_AB = sqrt((4-1)^2 + (4-0)^2) = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5
Length of AC:
A(1, 0) and C(13, 5)
Distance_AC = sqrt((13-1)^2 + (5-0)^2) = sqrt(12^2 + 5^2) = sqrt(144 + 25) = sqrt(169) = 13
The length of AC exceeds the length of AB by:
13 - 5 = 8 units.
2. Slope of BC:
To calculate the slope between two points (x1, y1) and (x2, y2), we use the slope formula:
Slope = (y2 - y1)/(x2 - x1)
Points B(4, 4) and C(13, 5):
Slope_BC = (5 - 4)/(13 - 4) = 1/9
Therefore, the slope of BC is 1/9.
1. To find the lengths of AC and AB, we can use the distance formula, which is based on the Pythagorean theorem. The distance formula states that the distance between two points (x1, y1) and (x2, y2) is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
For AC, we have the coordinates (1,0) and (13,5), so the length of AC can be found as follows:
AC = sqrt((13 - 1)^2 + (5 - 0)^2) = sqrt(12^2 + 5^2) = sqrt(144 + 25) = sqrt(169) = 13 units
For AB, we have the coordinates (1,0) and (4,4), so the length of AB can be found as follows:
AB = sqrt((4 - 1)^2 + (4 - 0)^2) = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5 units
Therefore, the length of AC exceeds the length of AB by 13 - 5 = 8 units.
2. The slope of a line passing through two points (x1, y1) and (x2, y2) can be found using the formula:
slope = (y2 - y1) / (x2 - x1)
For BC, we have the coordinates (4,4) and (13,5), so the slope of BC can be calculated as follows:
slope of BC = (5 - 4) / (13 - 4) = 1 / 9
Therefore, the slope of BC is 1/9.