A, B, C, and D are four points on the same straight line. The distances between the successive points are equal. If A is (1, -3), C is (4, a), and D is (b, 5). Find the values of a and b.
So, AC = 2AB = 2(D-C)
a-(-3) = 2(5-a)
2(b-4) = 4-1
Well, if A, B, C, and D are all on the same straight line and the distances between them are equal, that means they must be evenly spaced. So, if we know the coordinates of A and C, we can find the coordinates of B.
Since A is (1, -3) and C is (4, a), the x-coordinate of B would be right in the middle, which is (1 + 4)/2 = 2.5. The y-coordinate of B would also be in the middle of -3 and a, so it would be (-3 + a)/2.
Now, let's find the coordinates of D. Since D is on the same straight line as A, B, and C, its x-coordinate would be 3 units to the right of C's x-coordinate, which is 4. So, b would be 4 + 3 = 7. And since the distances between the points are equal, the y-coordinate of D would be 3 units higher than C's y-coordinate, which is a + 3.
Therefore, the values of a and b are a = -3 and b = 7.
By the way, did you hear about the mathematician who was afraid of negative numbers? He will stop at nothing to avoid them!
To find the values of "a" and "b", we can use the distance formula to determine the distance between points A and C, and points C and D. Since the distances between successive points are equal, we can set these two distances equal to each other.
Let's go step-by-step:
Step 1: Use the distance formula to find the distance between points A and C.
The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates of points A and C, we have:
d1 = √((4 - 1)^2 + (a - (-3))^2)
Simplifying the equation gives:
d1 = √(9 + (a + 3)^2)
d1 = √(9 + a^2 + 6a + 9)
d1 = √(a^2 + 6a + 18)
Step 2: Use the distance formula to find the distance between points C and D.
The distance formula is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Using the coordinates of points C and D, we have:
d2 = √((b - 4)^2 + (5 - a)^2)
Simplifying the equation gives:
d2 = √((b - 4)^2 + (25 - 10a + a^2))
Step 3: Since the distances between successive points are equal, we can set d1 equal to d2.
√(a^2 + 6a + 18) = √((b - 4)^2 + (25 - 10a + a^2))
Step 4: Square both sides of the equation to eliminate the square roots.
a^2 + 6a + 18 = (b - 4)^2 + (25 - 10a + a^2)
Step 5: Simplify and expand the equation.
a^2 + 6a + 18 = b^2 - 8b + 16 + 25 - 10a + a^2
Step 6: Rearrange the equation in terms of "a".
a^2 + 6a + 18 - a^2 + 10a = b^2 - 8b + 41
Step 7: Simplify.
16a + 18 = b^2 - 8b + 41
Step 8: Rearrange the equation in terms of "b".
b^2 - 8b - 16a = 41 - 18
Step 9: Combine like terms on the right side.
b^2 - 8b - 16a = 23
Now, we have an equation relating "a" and "b" that we can solve for their values.
To find the values of a and b, we can make use of the fact that the distances between consecutive points on the line are equal.
Let's start by finding the distance between A and C. We can use the distance formula to calculate this:
Distance between two points (x₁, y₁) and (x₂, y₂) = √((x₂ - x₁)² + (y₂ - y₁)²)
Using this formula, we have:
Distance AC = √((4 - 1)² + (a - (-3))²)
= √(3² + (a + 3)²)
= √(9 + (a + 3)²)
Next, let's find the distance between C and D:
Distance CD = √((b - 4)² + (5 - a)²)
Since the distances between the successive points are equal, we have the following equation:
Distance AC = Distance CD
So, we can equate the expressions we derived for AC and CD:
√(9 + (a + 3)²) = √((b - 4)² + (5 - a)²)
To solve for a and b, we need to isolate them in this equation. Squaring both sides of the equation will help eliminate the square roots.
(√(9 + (a + 3)²))² = (√((b - 4)² + (5 - a)²))²
(9 + (a + 3)²) = ((b - 4)² + (5 - a)²)
Expanding the squares:
9 + (a + 3)² = (b - 4)² + (5 - a)²
Simplifying:
9 + a² + 6a + 9 = b² - 8b + 16 + a² - 10a + 25
Combining like terms:
2a² - 16a + b² - 8b - 29 = 0
At this point, we have an equation with two variables, a and b. We'll need additional information or equations to solve for both variables simultaneously.