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Coordinate Geometry

Short Answer Type

301. Find the co-ordinates of the points which divide the line segment joining the points (-2, 0) and (0, 8) in four equal parts.

82 Views

302. Equilateral triangles are drawn on the sides of a right-angle triangle. Show that the area of the triangle on the hypotenuse is equal to the sum of the areas of triangles on the other two sides.

Given: A right-angle triangle right angled at B. Three equilateral triangles ABE, BCF and ACD are described on sides AB, BC and AC respectively.

To Prove: ar(∆ACD) = ar(∆ABE) + (∆BCF)
Proof: ∵ All triangle are equilateral therefore, by using AAA condition.

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When

increment ACD tilde increment ABE comma

then

fraction numerator ar left parenthesis increment ACD right parenthesis over denominator ar left parenthesis increment ABE right parenthesis end fraction equals AC squared over AB squared

rightwards double arrow

fraction numerator ar left parenthesis increment ABE right parenthesis over denominator ar left parenthesis increment ACD right parenthesis end fraction equals AB squared over AC squared space space space space space... left parenthesis straight i right parenthesis

[Taking reciprocal of both sides]
When

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, then

fraction numerator ar left parenthesis increment BCF right parenthesis over denominator ar left parenthesis increment ACD right parenthesis end fraction equals BC squared over AC squared

...(ii)
Adding (i) and (ii), we get

space space fraction numerator ar left parenthesis increment ABE right parenthesis over denominator ar left parenthesis increment ACD right parenthesis end fraction plus fraction numerator ar left parenthesis increment BCF right parenthesis over denominator ar left parenthesis increment ACD right parenthesis end fraction equals AB squared over AC squared plus BC squared over AC squared

rightwards double arrow

fraction numerator ar left parenthesis increment ABE right parenthesis plus ar left parenthesis increment BCF right parenthesis over denominator ar left parenthesis increment ACD right parenthesis end fraction equals fraction numerator AB squared plus BC squared over denominator AC squared end fraction

rightwards double arrow

space fraction numerator ar left parenthesis increment ABE right parenthesis plus ar left parenthesis increment BCF right parenthesis over denominator ar left parenthesis increment ACD right parenthesis end fraction equals AC squared over AC squared equals 1

rightwards double arrow

ar left parenthesis increment ABE right parenthesis plus ar left parenthesis increment BCF right parenthesis equals ar left parenthesis increment ACD right parenthesis

113 Views

303. Find the co-ordinates of the points equidistant from the points A(-2, -3), B(-1, 0) and C(7, -6)

86 Views

304. Two vertices of a ∆ABC are given by A(2, 3) and B(-2, 1) and its centroid is a

open parentheses 1 comma space 2 over 3 close parentheses

Find the co-ordinates of the third vertex C of the ∆ABC.

86 Views

305. Two vertices of a ∆ABC are given by A(6, 4) and B(-2, 2) and its centroid is C(3, 4). Find the coordinates of the third vertex C of the ∆ABC.

82 Views

306. Find the co-ordinates of the point equidistant from three given points A(5, 1), B(-3, -7) and C(7, -1).

141 Views

307. Find the value of p for which the points (-1, 3) (2, p) and (5, -1) are collinear.

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308. Find the co-ordinates of the point equidistant from three given points A(5, 3), B(5, -5) and C(1, -5).

304 Views

309. Find the value of p for which points (-5, 1), (1, p) and (4, -2) are collinear.

84 Views

310. In what ratio is the line segment joining the points (-2, -3) and (3, 7) divided by the y-axis? Also find the co-ordinates of the point of division.

101 Views