Coordinate Geometry

First quadrant, (4/3, 1)

Step 1: The intersection of medians (centroid) of a triangle with vertices (x1,y1), (x2,y2), (x3,y3) is at ((x1+x2+x3)/3, (y1+y2+y3)/3). Step 2: Vertices: A(0,0), B(4,0), C(0,3). Centroid x = (0+4+0)/3 = 4/3. Step 3: Centroid y = (0+0+3)/3 = 3/3 = 1. Step 4: So D = (4/3, 1). Since both coordinates are positive, D is in the first quadrant. Step 5: Option (2, 3/2) is wrong - this would be the midpoint of AC and BC, not the centroid. A common error is averaging only two vertices. Always use all three vertices for centroid.

x = 6, y = 3

Step 1: In a parallelogram, the diagonals bisect each other, meaning they share the same midpoint. Diagonals here are AC and BD. Step 2: Midpoint of AC = ((1+x)/2, (2+6)/2) = ((1+x)/2, 4). Step 3: Midpoint of BD = ((4+3)/2, (y+5)/2) = (7/2, (y+5)/2). Step 4: Setting midpoints equal: (1+x)/2 = 7/2, so 1+x = 7, x = 6. And 4 = (y+5)/2, so 8 = y+5, y = 3. Step 5: Therefore x = 6 and y = 3. Verify: Midpoint of AC = (7/2, 4) and Midpoint of BD = (7/2, 4). ✓ Common error: students try to equate adjacent sides rather than using the diagonal midpoint property.

(-5, 5)

Step 1: Distance of a point (a, b) from the x-axis = |b| and distance from the y-axis = |a|. Step 2: Equidistant from both axes means |a| = |b| = 5. So a = ±5 and b = ±5. Step 3: The point is in the second quadrant, where x-coordinate is negative and y-coordinate is positive. Step 4: Therefore a = -5 and b = +5. So the coordinates are (-5, 5). Step 5: Verify: Distance from x-axis = |5| = 5 ✓. Distance from y-axis = |-5| = 5 ✓. Point (-5, 5) is in QII since x < 0 and y > 0 ✓. Options (5,5) is QI and (-5,-5) is QIII - wrong quadrants.

(3, -4)

Step 1: Start with P = (-3, 4). Step 2: Reflecting across the x-axis: the x-coordinate stays the same, but the y-coordinate changes sign. So P' = (-3, -4). Step 3: Reflecting P' across the y-axis: the y-coordinate stays the same, but the x-coordinate changes sign. So P'' = (3, -4). Step 4: P'' = (3, -4) lies in the fourth quadrant (positive x, negative y). Step 5: Note: reflecting across both axes is equivalent to a 180° rotation about the origin, which changes signs of both coordinates: (-3, 4) → (3, -4). Common errors include swapping which coordinate changes sign for each reflection.