Verify that points P(-2, 2), Q(2, 2) and R(2, 7) are vertices of a right angled triangle.
Given P(-2, 2), Q(2, 2) and R(2, 7).If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.By distance formula , distance between two points = √[(x2-x1)2+(y2-y1)2]PQ = √[(2-(-2))2+(2-2)2]PQ = √[(42+02]PQ = √16PQ […]
Find the point on the X-axis which is equidistant from A(-3, 4) and B(1, -4).
Let C be the point on X axis equidistant from A(-3,4) and B(1,-4).Since C lies on X axis, the Y co-ordinate of C is 0.Let C = (x,0)Co-ordinates of A = (-3, 4)Co-ordinates of B = (1, -4)Since C is equidistant from A and B ,AC = BCBy distance formula, d(A,C) = √[(x2-x1)2+(y2-y1)2]d(A,C) = √[(x-(-3))2+(0-4)2]d(A,C) […]
Determine whether the points are collinear.
(1) If the sum of any two distances out of d(A, B), d(B, C) and d(A, C) is equal to the third , then the three points A, B and C are collinear.we will find d(A, B), d(B, C) and d(A, C).Co-ordinates of A = (1,-3)Co-ordinates of B = (2,-5)Co-ordinates of C = (-4,7)By distance […]
Find the distance between each of the following pairs of points.
(1) Let A(x1, y1) and B(x2 , y2) be the given pointsBy distance formula d(A,B) = √[(x2-x1)2+(y2-y1)2]Here x1 = 2, y1 = 3 , x2 = 4, y2 = 1d(A,B) = √[(4-2)2+(1-3)2]= √[22+(-2)2]= √8= 2√2Hence the distance between A and B is 2√2 units. (2) Let P(x1, y1) and Q(x2 , y2) be the given pointsBy distance formula d(P,Q) = √[(x2-x1)2+(y2-y1)2]Here […]
Draw a circle of diameter 6.4 cm. Take a point R at a distance equal to its diameter from the centre. Draw tangents from point R.
Construction steps:1.Draw a circle of radius = 6.4/ 2 = 3.2 cm with centre O.2. Mark a point R in the exterior of the circle so that OR = 6.4 cm.3. Draw segment OR. Draw a perpendicular bisector of OR. Mark its midpoint M.4. Draw a circle with radius OM and centre M.5. Mark the point of intersection of the […]
Draw any circle. Take any point A on it and construct tangent at A without using the centre of the circle.
Analysis: As shown in the above figure, let line l be the tangent to the circle at point A. Line AB is a chord andBCA is an inscribed angle. Now by tangent- secant angle theorem, BCA BAR.By converse of tangent- secant theorem, if we draw the line l such that, BAR BCA, then it will be the […]
Draw a circle with centre O and radius 3.5 cm. Take point P at a distance 5.7 cm from the centre. Draw tangents to the circle from point P.
Construction steps.1. Draw a circle of radius 3.5 cm with centre O.2. Mark a point P in the exterior of the circle so that OP = 5.7 cm3.Join OP. Draw perpendicular bisector of segment OP and mark the midpoint M.4. Draw a circle with radius OM and centre M.5.Name the point of intersection of the two […]
Select the correct alternative for each of the following questions.
(1) The number of tangents that can be drawn to a circle at a point on the circle is …………… .(A) 3 (B) 2 (C) 1 (D) 0Solution:The number of tangents that can be drawn to a circle at a point on the circle is 1.Hence option C is the answer. (2) The maximum number of […]
Draw a circle of radius 3.3 cm Draw a chord PQ of length 6.6 cm. Draw tangents to the circle at points P and Q. Write your observation about the tangents.
Rough figure is given below. Since tangent is perpendicular to radius, OP line l .Also OQ line m.The perpendicular line segments to OP and OQ at point P and Q will give the required tangents at P and Q.Given radius = 3.3 cm.Diameter = 3.3 ×2 = 6.6cm.So chord PQ is the diameter of the circle.The […]
Draw a circle of radius 3.6 cm. Draw a tangent to the circle at any point on it without using the centre.
Analysis: As shown in the above figure, let line l be the tangent to the circle at point C. Line CB is a chord and CAB is an inscribed angle. Now by tangent- secant angle theorem, CAB BCD.By converse of tangent- secant theorem, if we draw the line CD such that, CAB BCD, then it […]