1. Find the distance between the following pairs of points:
(i) (2,3), (4,1)
(ii) (−5,7), (−1,3)
(iii) (a, b), (-a, -b)
Get the complete, step-by-step math solution for: "1. Find the distance between the following pairs of points: (i) (2, 3), (4, 1) (ii) (-5, 7), (-1, 3) (iii) (a, b), (-a, -b)". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Recall the Distance Formula
The distance formula is used to calculate the distance d between two points (x1,y1) and (x2,y2) in a Cartesian coordinate system. It is derived from the Pythagorean theorem.
Step 2: Calculate Distance for (i) (2, 3) and (4, 1)
For the first pair of points, (2,3) and (4,1), we substitute x1=2, y1=3, x2=4, and y2=1 into the distance formula. After performing the subtractions and squaring the results, we add them and take the square root. The distance simplifies to 22 units.
Step 3: Calculate Distance for (ii) (-5, 7) and (-1, 3)
For the second pair of points, (−5,7) and (−1,3), we substitute x1=−5, y1=7, x2=−1, and y2=3 into the distance formula. We then simplify the expression to find the distance, which is 42 units.
Step 4: Calculate Distance for (iii) (a, b) and (-a, -b)
For the third pair of points, (a, b) and (-a, -b), we substitute x1=a, y1=b, x2=−a, and y2=−b into the distance formula. We simplify the algebraic expression by combining like terms, squaring them, and factoring out common terms. The final distance is 2a2+b2 units.