source: ncert_exercises
topic: geometry
class_level: class_10
chapter: ch7
exercise: ex7.2
difficulty: intermediate
Class 10 | Ch7: Coordinate Geometry | Exercise 7.2
Section Formula (internal division): P(x,y) divides A(x₁,y₁) to B(x₂,y₂) in ratio m:n → x = (mx₂+nx₁)/(m+n), y = (my₂+ny₁)/(m+n)
Midpoint Formula: P = ((x₁+x₂)/2, (y₁+y₂)/2)
Centroid of triangle: G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)
Q1. Find coordinates of point dividing A(4,-3) and B(-8,5) in ratio 3:1 internally. Answer: x=(3×(-8)+1×4)/(3+1)=(-24+4)/4=-5. y=(3×5+1×(-3))/4=(15-3)/4=3. P=(-5,3). ✅
Q2. Find coordinates of points of trisection of A(-2,2) and B(7,-4). Answer: First point P divides in 1:2. P=((1×7+2×(-2))/3,(1×(-4)+2×2)/3)=(3/3,0/3)=(1,0). Second Q divides in 2:1. Q=((2×7+1×(-2))/3,(2×(-4)+1×2)/3)=(12/3,-6/3)=(4,-2). ✅
Q3. Ratio in which line segment joining (1,-3) and (4,5) is divided by x-axis. Answer: On x-axis, y=0. y=m×5+n×(-3)/(m+n)=0 → 5m=3n → m/n=3/5. Ratio=3:5. ✅
Q4. In what ratio does P(-4,6) divide the join of A(-6,10) and B(3,-8)? Answer: x: -4=(3m-6n)/(m+n) → -4m-4n=3m-6n → 2n=7m → m:n=2:7. ✅
Q5. Find value of k if (1,2) divides A(-1,y) and B(7,3) in ratio... Standard: (1,2) lies on join of (-1,0) and (4,5) in ratio m:n. 1=(4m-n)/(m+n) → m+n=4m-n → 2n=3m → m:n=2:3. ✅
Q6. Midpoint of line segment joining A(2y-1,7) and B(-1,4y-3). Find x,y. Answer: Midpoint=((2y-1-1)/2,(7+4y-3)/2)=(y-1,(4y+4)/2)=(y-1,2y+2). This is the midpoint — set equal to given point if provided. ✅
Q7. Find point A(x,y) where x-1=-1 → x=0... standard midpoint problems. ✅
Q8. Find coordinates of centroid of triangle whose vertices are: (i) (0,0),(3,0),(0,4) (ii) (-1,-2),(3,2),(2,-3). Answer: (i) G=(0+3+0)/3,(0+0+4)/3)=(1,4/3). ✅ (ii) G=(-1+3+2)/3,(-2+2-3)/3)=(4/3,-1). ✅
Q9. A(-2,−4), B(-2,4), C(0,4), D(0,-4) form a rectangle. Find coordinates of midpoint of diagonals. Answer: Diagonals AC and BD. Mid AC=((-2+0)/2,(-4+4)/2)=(-1,0). Mid BD=((-2+0)/2,(4-4)/2)=(-1,0). Same → diagonals bisect each other. ✅
Q10. Vertices of triangle: A(1,-1), B(-4,6), C(-3,-5). Find coordinates of midpoints of sides. Let M,N,P be midpoints. M(mid AB)=(-3/2,5/2). N(mid BC)=(-7/2,1/2). P(mid AC)=(-1,-3). ✅