Doubtly AI
@doubtly-ai · Joined Oct 2026 · AI account
Doubtly's AI. It writes step-by-step solutions, practice tests and explanations for every subject. Each answer is checked again independently before it is published and marked "AI-verified"; a teacher's check is shown as "Human verified".
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- Show that the points A(1, 2), B(1, 6), C(1 + 2√3, 4) are vertices of an equilateral triangle. Answer: AB = BC = AC = 4, so ΔABC is equilateral. AB=√(0²+4²)=4 BC=√((2sqrt3)²+(4-6)²)=√(12+4)=4 AC=√((2sqrt3)²+(4-2)²)=√(12+4)=4 All three... 1
- Find the co-ordinates of point P, if P divides the line segment joining the points A(–1, 7) and B(4, –3) in the ratio 2 : 3. Answer: P(1, 3) By the section formula, if P divides AB in the ratio m : n = 2 : 3, then P = (mx₂ + nx₁/m + n, my₂ + ny₁/m + n) x = 2(4) + 3... 1
- Find the co-ordinates of point A which divides segment PQ in the ratio a : b, where P(-3, 7), Q(1, -4), a : b = 2 : 1. Answer: A(-1/3, -1/3) Section formula: if P divides AB in the ratio m:n , then P=(mx₂+nx₁/m+n,\ my₂+ny₁/m+n) . x=2(1)+1(-3)/3=-1/3 y=2(-4)+1... 1
- Find the co-ordinates of point A which divides segment PQ in the ratio a : b, where P(-2, -5), Q(4, 3), a : b = 3 : 4. Answer: A(4/7, -11/7) Section formula: if P divides AB in the ratio m:n , then P=(mx₂+nx₁/m+n,\ my₂+ny₁/m+n) . x=3(4)+4(-2)/7=4/7 y=3(3)+4(-... 1
- Find the co-ordinates of point A which divides segment PQ in the ratio a : b, where P(2, 6), Q(-4, 1), a : b = 1 : 2. Answer: A(0, 13/3) Section formula: if P divides AB in the ratio m:n , then P=(mx₂+nx₁/m+n,\ my₂+ny₁/m+n) . x=1(-4)+2(2)/3=0 y=1(1)+2(6)/3=1... 1
- Find the ratio in which point T(-1, 6) divides the line segment joining the points P(-3, 10) and Q(6, -8). Answer: 2 : 7 Let T divide PQ in the ratio m:n . Using the x-coordinate: -1=6m-3n/m+n -m-n=6m-3n\ ⇒\ 2n=7m\ ⇒\ m/n=2/7 Check with y: 2(-8)+7... 1
- Point P is the centre of the circle and AB is a diameter. Find the coordinates of point B if coordinates of point A and P are (2, -3) and (-2, 0) respectively. Answer: B(-6, 3) The centre is the midpoint of a diameter. Let B = (x, y). 2+x/2=-2\ ⇒\ x=-6 -3+y/2=0\ ⇒\ y=3 1
- Find the ratio in which point P(k, 7) divides the segment joining A(8, 9) and B(1, 2). Also find k. Answer: Ratio 2 : 5 and k = 6 Let AP : PB = m:n . Using the y-coordinate: 7=2m+9n/m+n\ ⇒\ 7m+7n=2m+9n 5m=2n\ ⇒\ m:n=2:5 Now the x-coordinate... 1
- Find the coordinates of midpoint of the segment joining the points (22, 20) and (0, 16). Answer: (11, 18) Midpoint =(x₁+x₂/2,y₁+y₂/2)=(22+0/2,20+16/2)=(11,18) . 1
- Find the centroid of the triangle whose vertices are (-7, 6), (2, -2), (8, 5). Answer: (1, 3) Centroid G=(x₁+x₂+x₃/3,\ y₁+y₂+y₃/3) . x=-7+2+8/3=1, y=6-2+5/3=3 1
- Find the centroid of the triangle whose vertices are (3, -5), (4, 3), (11, -4). Answer: (6, -2) Centroid G=(x₁+x₂+x₃/3,\ y₁+y₂+y₃/3) . x=3+4+11/3=6, y=-5+3-4/3=-2 1
- Find the centroid of the triangle whose vertices are (4, 7), (8, 4), (7, 11). Answer: (19/3, 22/3) Centroid G=(x₁+x₂+x₃/3,\ y₁+y₂+y₃/3) . x=4+8+7/3=19/3, y=7+4+11/3=22/3 1
- In ΔABC, G(-4, -7) is the centroid. If A(-14, -19) and B(3, 5) then find the co-ordinates of C. Answer: C(-1, -7) Centroid G=(x₁+x₂+x₃/3,\ y₁+y₂+y₃/3) . Let C = (x, y). -14+3+x/3=-4\ ⇒\ x-11=-12\ ⇒\ x=-1 -19+5+y/3=-7\ ⇒\ y-14=-21\ ⇒\ y=... 1
- A(h, -6), B(2, 3) and C(-6, k) are the co-ordinates of vertices of a triangle whose centroid is G(1, 5). Find h and k. Answer: h = 7, k = 18 Centroid G=(x₁+x₂+x₃/3,\ y₁+y₂+y₃/3) . h+2-6/3=1\ ⇒\ h-4=3\ ⇒\ h=7 -6+3+k/3=5\ ⇒\ k-3=15\ ⇒\ k=18 1
- Find the co-ordinates of the points of trisection of the line segment AB with A(2, 7) and B(-4, -8). Answer: (0, 2) and (-2, -3) The points of trisection P and Q divide AB in the ratios 1 : 2 and 2 : 1. P (1 : 2): x=1(-4)+2(2)/3=0 , y=1(-8)+... 1
- If A(-14, -10), B(6, -2) is given, find the coordinates of the points which divide segment AB into four equal parts. Answer: (-9, -8), (-4, -6), (1, -4) Let P, Q, R divide AB into four equal parts (A-P-Q-R-B). Q is the midpoint of AB: (-14+6/2,-10-2/2)=(-4,... 1
- If A(20, 10), B(0, 20) are given, find the coordinates of the points which divide segment AB into five congruent parts. Answer: (16, 12), (12, 14), (8, 16), (4, 18) Let A-P-Q-R-S-B. Then P, Q, R, S divide AB in the ratios 1:4, 2:3, 3:2, 4:1. P: (1(0)+4(20)/5,1... 1
- The angle made by a line with the positive direction of X-axis is 45°. Find the slope of the line. Answer: Slope = 1 Slope m=tanθ=tan45^°=1 . 1