The points A(1,5) and B(9,3) are part of a triangle ΔABC. The triangle has a right angle at A and and sides satisfy AB=2AC. Find a point C such that C lies above the line AB.
Did you make your sketch ?
Two concepts are needed here:
1. the slope of perpendicular lines are negative reciprocals of each other
2. You have to know how to find the distance between two points.
1.
slope of AB = (9-1)/(3-5) = -4
so slope of AC = + 1/4
but slope of AC = (y-5)/(x-1)
so:
(y-5)/(x-1) = 1/4
x-1 = 4y - 20
x = 4y - 19
2.
AB = √68 , leaving it up to you to check that.
AC = √((x-1)^2 + (y-5)^2 )
but AB = 2AC
√68 = 2√((x-1)^2 + (y-5)^2 )
square both sides and expand
68 = x^2 - 2x + 1 + y^2 - 10y + 25
x^2 - 2x + y^2 - 10y -42 = 0
subbing in x = 4y-19
(4y-19)^2 - 2(4y-19) + y^2 - 10y - 42 = 0
leaving it up to you to solve for y, then find the matching x
you will get 2 points, but you want the one above line AB
check my arithmetic
To find point C such that it lies above the line AB, we can follow these steps:
1. Determine the length of AB.
- Use the distance formula to calculate the distance between points A(1,5) and B(9,3).
- Distance formula: d = √((x₂ - x₁)² + (y₂ - y₁)²)
- Substitute the coordinates of A and B into the formula:
d(AB) = √((9 - 1)² + (3 - 5)²)
= √(8² + (-2)²)
= √(64 + 4)
= √68
≈ 8.246
2. Determine the length of AC.
- Since AB = 2AC, we can equate the lengths:
AB = 2AC
8.246 = 2AC
AC = 8.246/2
≈ 4.123
3. Determine the slope of AB.
- Use the formula for slope to calculate the slope between points A and B.
- Slope formula: m = (y₂ - y₁)/(x₂ - x₁)
- Substitute the coordinates of A and B into the formula:
m(AB) = (3 - 5)/(9 - 1)
= (-2)/8
= -1/4
4. Determine the slope of a line perpendicular to AB.
- The negative reciprocal of the slope of AB will give us the slope of a line perpendicular to AB.
- Slope of perpendicular line = -1/m(AB)
- Substitute the slope of AB into the formula:
m(perpendicular) = -1/(-1/4)
= 4
5. Determine point C.
- Since C lies above the line AB, its y-coordinate will be greater than the y-coordinate of A.
- Given that A(1,5), we can choose a value for the x-coordinate of C, and using the slope of the perpendicular line, we can find the corresponding y-coordinate.
- Set the x-coordinate of C as x.
- Slope-intercept form of a line equation: y = mx + b
- Substitute the slope and coordinates of A into the equation:
5 = 4(1) + b
5 = 4 + b
b = 5 - 4
= 1
- Therefore, the equation of the line is y = 4x + 1.
- Substitute the chosen x-coordinate into the equation to find the y-coordinate of C.
- For example, let's choose x = 3:
y = 4(3) + 1
= 12 + 1
= 13
- Therefore, C(3,13) lies above the line AB.
In summary, point C is at (3, 13) such that it lies above the line AB, given that A(1,5) and B(9,3) are part of a triangle ΔABC, and the triangle has a right angle at A, and the sides satisfy AB = 2AC.