Centroid and Incenter of a Triangle
Category : 9th Class
The coordinate of centroid of a triangle whose vertices are \[({{x}_{1}},{{y}_{1}}),({{x}_{2}},{{y}_{2}})\] and \[({{x}_{3}},{{y}_{3}})\] is \[\left( \frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\frac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3} \right)\]
The coordinate of the in centre of a triangle ABC whose vertices are \[A({{x}_{1}},{{y}_{1}}),B({{x}_{2}},{{y}_{2}})\] and \[C({{x}_{3}},{{y}_{3}})\] is \[\left( \frac{a{{x}_{1}}+b{{x}_{2}}+c{{x}_{3}}}{a+b+c},\frac{a{{y}_{1}}+b{{y}_{2}}+c{{y}_{3}}}{a+b+c} \right)\]
The ratio in which line \[y=x-2\] divides the line segment joining (8, 9) and (3,-1) is_____.
(a) 3:2
(b) \[\frac{3}{4}:\frac{4}{2}\]
(c) 2:3
(d) 3:3
(e) None of these
Answer: (c)
Explanation:
Given line is \[y=x-2\] ..... (i)
Let P=(8,9) and Q=(3,-1)
let line (i) divides PQ, in the ratio K : 1 at point R;
then \[R=\left( \frac{8k+3}{k+1},\frac{9k-1}{k+1} \right)\]
Since R lies on \[y=x-2\], therefore,
\[\frac{9k-1}{k+1}=\frac{8k+3}{k+1}-2\]
\[\Rightarrow \]\[\frac{9k-1}{k+1}-\frac{8k+3}{k+1}=-2\] \[\Rightarrow \]\[\frac{9k-1-8k-3}{k+1}=-2\]
\[\Rightarrow \]\[\frac{k-4}{k+1}=-2\] \[\Rightarrow \]\[k-4=-2(k+1)\] \[\Rightarrow \]\[k-4=-2k-2\] \[\Rightarrow \]\[3k=4-2\] \[\Rightarrow \]\[k=\frac{2}{3}\]
Then the ratio is k : 1 \[\Rightarrow \]\[\frac{2}{3}:1\] \[\Rightarrow \]\[2:3\]
Find the coordinate of point which trisect the line segment joining the points (1, 3) and (3, 9).
(a) (5, 5), (3, 3)
(b) (5, 5), (7, 7)
(c) \[\left( \frac{5}{3},5 \right)\left( \frac{7}{3},7 \right)\]
(d) \[\left( \frac{5}{3},\frac{7}{3} \right)\left( 5,7 \right)\]
(e) None of these
Answer: (c)
Explanation:
The coordinate of point P is
\[x=\frac{3\times 1+2\times 1}{3}=\frac{5}{3}\]
\[y=\frac{3\times 2+9\times 1}{3}=\frac{{{\bcancel{15}}^{3}}}{3}=5\]
The coordinate of point Q is
\[x=\frac{1\times 1+3\times 2}{3}=\frac{7}{3}\]
\[y=\frac{3\times 1+9\times 2}{3}=\frac{21}{3}=7\]
Thus the coordinate of point P is \[\left( \frac{5}{3},5 \right)\] and Q is \[\left( \frac{7}{3},7 \right)\].
The coordinate of centroid of a triangle whose vertices are (3, 2), (-3,-1) and (0, -1) is.....
(a) (0, 0)
(b) (0, 3)
(c) (3, 0)
(d) (0,-5)
(e) None of these
Answer: (a)
If the point (m, n) is equilateral from \[(x+y,y-x)\] and \[(x-y,x+y)\] then which one of the following options is true?
(a) \[\frac{x+y}{x-y}=\frac{m+n}{n-m}\]
(b) \[(m+n)(m-n)=(x+y)(x-y)\]
(c) \[(m+n)(x+y)=(m-n)(x-y)\]
(d) \[\frac{n-m}{n+m}=\frac{x-y}{x+y}\]
(e) None of these
Answer: (a)
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