(ii) 
(i) $\mathrm{A}(5,-12)$ Let $A(0,0)$ be the origin $$ &A=\sqrt{(5-0)^{2}+(-12-0)^{2}}\\ $$ $$ \begin{aligned} &=\sqrt{(5)^{2}+(-12)^{2}} \\ &=\sqrt{25+144} \\ &=\sqrt{169} \\ &=13 \text { units }...
and 
(iv) 
and 
(iii) $\mathrm{A}(-6,-4)$ and $\mathrm{B}(9,-12)$ points are $\mathrm{A}(-6,-4)$ and $\mathrm{B}(9,-12)$ => $\left(x_{1}=-6, y_{1}=-4\right)$ and $\left(x_{2}=9, y_{2}=-12\right)$...
and 
(ii) 
and 
(i) $\quad \mathrm{A}(9,3)$ and $\mathrm{B}(15,11)$ The given points are $A(9,3)$ and $B(15,11)$. Then $\left(x_{1}=9, y_{1}=3\right)$ and $\left(x_{2}=15, y_{2}=11\right)$ $A...
Solution: The statement given in the question is false. Justification: If the area of a triangle formed by its points equals 0, thenΒ the points are collinear. Provided, ${{x}_{1}}~=\text{ }0,\text{...
Solution: The statement given in the question is true. Justification: Equation of the line conatining the points A and B using the two-point form is, $\frac{y-6}{x+4}=\frac{-6-6}{-4+4}$...
Solution: The statement given in the question is true. Justification: Distance formula, $d~=\text{ }\surd \text{ }({{({{x}_{2}}~\text{ }{{x}_{1}})}^{2}}~+\text{ }{{({{y}_{2}}~\text{...
Solution: The side AB has P as a mid-point, Coordinate of P = ( (-1 β 1)/2, (-1 + 4)/2 ) = (-1, 3/2) In the same way, Q, R, and S are (As Q is mid-point of BC, R is midpoint of CD and S is...
, B 
and C 
are the vertices of triangle ABC, find the coordinates of the centroid of the triangle.Solution: (i) If $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right)$ and $C\left(x_{3}, y_{3}\right)$ are the vertices of triangle $A B C$, the coordinates of centroid can be given as...
(i) The median from A meets BC at D. Find the coordinates of point D. (ii) Find the coordinates of the point P on AD such that AP : PD = 2 : 1. Solution: (i) D's coordinates can be computed using...
Given that: A (4, 6), B (1, 5) and C (7, 2) are the vertices of a βABC. AD/AB = AE/AC = 1/4 AD/(AD + BD) = AE/(AE + EC) = 1/4 Points D and E divide AB and AC in a 1:3 ratio, respectively. D's...
(i) Taking A as origin, find the coordinates of the vertices of the triangle. (ii) What will be the coordinates of the vertices of triangle PQR if C is the origin?Also calculate the areas of the...
Solution: Let ABCD be a square with A(-1,2) and B(-1,2) (3,2). And Point O is the point where AC and BD intersect. Now find the coordinates of the points B and D. Step 1: Find coordinates of point O...
Solution: The points on a circle are A = (6, -6), B = (3, -7), and C = (3, 3). OA = OB = OC if O is the centre. (radii are equal) If O is equal to (x, y), then OA = β[(x β 6)2Β + (y + 6)2] OB =Β β[(x...
Solution: The area of the triangle formed by the given points must be 0 if the points are collinear. Let the vertices of a triangle are (x, y), (1, 2) and (7, 0), The area of a triangle =Β 1/2 Γ...
Solution: Consider how line 2x + y β 4 = 0 divides line AB in a k:1 ratio joinedΒ by the two points A(2, -2) and B(3, 7). The following are the coordinates of the point of division: x = (2 + 3k)/(k +...
Solution: Let A (4, -6), B (3, -2), and C (5, 2) be the vertices of the triangle. Let the side BC of triangle ABC has a midpoint as D. As a result, in ΞABC, AD is the median. Midpoint of BC =...
(-4, -2), (-3, -5), (3, -2) and (2, 3). Solution: Let the quadrilateral's vertices be A (- 4, β 2), B (β 3, β 5), C (3, β 2), and D (2, 3). Divide the quadrilateral into two triangles by...
Solution: Let the triangle's vertices be A (0, -1), B (2, 1), and C (3, 1). (0, 3). Let D, E, and F be the midpoints of the triangle's sides. D, E, and F coordinates are given by D .= (0+2/2, -1+1/2...
(i) (7, -2), (5, 1), (3, -k) (ii) (8, 1), (k, -4), (2, -5) Solution: (i) The area of the triangle formed by collinear points is always zero. Let the vertices of a triangle be (7, -2) (5, 1), and (3,...
(i) (2, 3), (-1, 0), (2, -4)<br> (ii) (-5, -1), (3, -5), (5, 2) Solution: The formula for calculating the area of a triangle is = 1/2 Γ [x1(y2Β β y3) + x2(y3Β β y1) + x3(y1Β β y2)] (i) Given here,...
Solution: The vertices of a rhombus ABCD are A(3, 0), B (4, 5), C(β 1, 4) and D (β 2, β 1) Length of diagonal AC$=\sqrt{{{\left( 3-\left( -1 \right) \right)}^{2}}+{{\left( 0-4...
Solution: Make a figure, line dividing by 4 points. The points X, Y, and Z divide the line segment in the ratios 1:3, 1:1, and 3:1, respectively, as shown in the diagram. Coordinates of X$=\left(...
Solution: Points A and B have coordinates of (-2,-2) and (2,-4) respectively. Because AP = 3/7, AB As a result, AP: PB = 3:4 The line segment AB is divided in 3:4 by point P. Coordinate of P...
Solution: Let's say point A's coordinates are (x, y). The midpoint of AB is (2, β 3), which is the circle's centre. Coordinate of B = (1, 4) (2, -3) =((x+1)/2 , (y+4)/2) (x+1)/2 = 2 and (y+4)/2 = -3...
Solution: Let A,B,C and D be the points of a parallelogram as A(1, 2), B(4, y), C(x, 6) and D(3, 5). The midpoint of a parallelogram is the same since the diagonals bisect each other. To find...
Solution: Let k : 1 be the ratio of the line segment connecting A (1, β 5) and B ( β 4, 5) divided by the x-axis. As a result, the coordinates of the division point, P(x, y), are ((-4k+1)/(k+1),...
Solution: Consider the ratio k :1 which connects the line segment ( -3, 10) and (6, -8) to the point ( -1, 6). As a result, Β -1 = ( 6k-3)/(k+1) βk β 1 = 6k -3 7k = 2...
We concluded from the given instructions that Niharika placed the green flag at 1/4th of the distance AD, or (1/4 Γ100) m = 25 metres from the 2nd line's starting point. As a result, this...
Solution: Let P (x1, y1) and Q (x2, y2) be the trisection points of the line segment joining the provided points, i.e. AP = PQ = QB. As a result, point P internally divides AB in the ratio 1:2....
Solution: Let the required point is P(x, y). Calculating usingΒ the section formula, we will get: x = (2Γ4 + 3Γ(-1))/(2 + 3) = (8 β 3)/5 = 1 y = (2Γ-3 + 3Γ7)/(2 + 3) = (-6 + 21)/5 = 3 As a result,...
Solution: (3, 6) and ( β 3, 4) are both equidistant from point (x, y). $\sqrt{{{\left( x-3 \right)}^{2}}+{{\left( y-6 \right)}^{2}}}=\sqrt{{{\left( x-\left( -3 \right) \right)}^{2}}+{{\left( y-4...
Solution: Given: PQ = QR because Q (0, 1) is equidistant between P (5, β 3) and R (x, 6). Step 1: Using the distance formula, calculate the distance between PQ and QR. PQ$=\sqrt{{{\left( 5-0...
Solution: Given: There is a distance of 10 between (2, β 3) and (10, y). Using the formula for distance, PQ$=\sqrt{{{\left( 10-2 \right)}^{2}}+{{\left( y+3 \right)}^{2}}}=\sqrt{{{\left( 8...
Solution: To locate a point on the x-axis. As a result, its y-coordinate will be zero. Let's say the x-axis point is (x,0). Consider the following : A = (x, 0), B = (2, β 5), and C = (- 2, 9)....
Solution: Let the vertices A, B, C, and D of the given quadrilateral be represented by the points (4, 5), (7, 6), (4, 3), and (1, 2), respectively. AB$=\sqrt{{{\left( 7-4 \right)}^{2}}+{{\left( 6-5...
(i) (- 1, β 2), (1, 0), (- 1, 2), (- 3, 0) (ii) (- 3, 5), (3, 1), (0, 3), (- 1, β 4) Solution: (i) Let the points (- 1, β 2), (1, 0), ( β 1, 2), and ( β 3, 0) represent the quadrilateral's vertices...
Solution: The coordinates of points A, B, C, and D are (3, 4), (6, 7), (9, 4) and (6, 1), respectively, as shown in the figure. Using the distance formula, we can calculate the distance between two...
Solution: Any isosceles triangle has two sides that are equal. We'll find the distance between all of the points to see if they're vertices of an isosceles triangle. Let the vertices A, B, and C be...
Solution: All three points are collinear if the total of the lengths of any two line segments equals the length of the third line segment. Consider the following values, A = (1, 5) B = (2, 3)...
Solution: Consider the case of town A at point (0, 0). As a result, town B will be at the point (36, 15). The distance between the points (0, 0) and (36, 15) $d=\sqrt{{{(36-0)}^{2}}+{{\left( 15-0...
(i) (a, b), (- a, β b) Solution: (i) Distance between (a, b), (-a, -b) $d=\sqrt{{{\left( -a-a \right)}^{2}}+{{\left( -b-b \right)}^{2}}}=\sqrt{{{\left( -2a \right)}^{2}}+{{\left( -2b \right)}^{2}}}$...
(i) (2, 3), (4, 1) (ii) (-5, 7), (-1, 3) Solutions: The distance formula, say d, is used to calculate the distance between two points (x1, y1) and (x2, y2), $d=\sqrt{{{\left( {{x}_{2}}-{{x}_{1}}...
We know that in an isosceles triangle, length of two sides is equal. We know that formula to find the distance (d) between two points $\left( {{x}_{1}},{{y}_{1}} \right)$and $\left(...
All three points are collinear if the total of the lengths of any two line segments equals the length of the third line segment. Consider the following values, A = (1, 5) B = (2, 3) and C = (-2,...