1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

(i) If the vertices of $∆ABC$ be A(1, −3), B(4, p) and C(−9, 7) and its area is 15 square units, find the values of p. [CBSE 2012] (ii) The area of a triangle is 5 sq units. Two of its vertices are (2, 1) and (3, –2). If the third vertex is $\left(\frac{7}{2},y\right)$, find the value of y. [CBSE 2017]

Open in App
Solution

(i) Let $A\left({x}_{1},{y}_{1}\right)=A\left(1,-3\right),B\left({x}_{2},{y}_{2}\right)=B\left(4,p\right)\mathrm{and}C\left({x}_{3},{y}_{3}\right)=C\left(-9,7\right)$. Now $\mathrm{Area}\left(∆ABC\right)=\frac{1}{2}\left[{x}_{1}\left({y}_{2}-{y}_{3}\right)+{x}_{2}\left({y}_{3}-{y}_{1}\right)+{x}_{3}\left({y}_{\mathit{1}}\mathit{-}{y}_{\mathit{2}}\right)\right]\phantom{\rule{0ex}{0ex}}⇒15=\frac{1}{2}\left[1\left(p-7\right)+4\left(7+3\right)-9\left(-3-p\right)\right]\phantom{\rule{0ex}{0ex}}⇒15=\frac{1}{2}\left[10p+60\right]\phantom{\rule{0ex}{0ex}}⇒\left|10p+60\right|=30$ Therefore $⇒10p+60=-30\mathrm{or}30\phantom{\rule{0ex}{0ex}}⇒10p=-90\mathrm{or}-30\phantom{\rule{0ex}{0ex}}⇒p=-9\mathrm{or}\mathit{}\mathit{-}3$ Hence, $p=-9\mathrm{or}p=-3$. (ii) Let $A\left({x}_{1},{y}_{1}\right)=A\left(2,1\right),B\left({x}_{2},{y}_{2}\right)=B\left(3,-2\right)\mathrm{and}C\left({x}_{3},{y}_{3}\right)=C\left(\frac{7}{2},y\right).$ Now $\mathrm{Area}\left(∆ABC\right)=\frac{1}{2}\left|{x}_{1}\left({y}_{2}-{y}_{3}\right)+{x}_{2}\left({y}_{3}-{y}_{1}\right)+{x}_{3}\left({y}_{1}\mathit{-}{y}_{\mathit{2}}\right)\right|\phantom{\rule{0ex}{0ex}}⇒5=\frac{1}{2}\left|2\left(-2-y\right)+3\left(y-1\right)+\frac{7}{2}\left(1+2\right)\right|\phantom{\rule{0ex}{0ex}}⇒10=\left|-4-2y+3y-3+\frac{21}{2}\right|\phantom{\rule{0ex}{0ex}}⇒10=\left|y+\frac{7}{2}\right|\phantom{\rule{0ex}{0ex}}⇒10=y+\frac{7}{2}\mathrm{or}-10=y+\frac{7}{2}\phantom{\rule{0ex}{0ex}}⇒y=\frac{13}{2}\mathrm{or}y=\frac{-27}{2}$ Hence, y = $\frac{13}{2}\mathrm{or}\frac{-27}{2}.$

Suggest Corrections
0
Join BYJU'S Learning Program
Related Videos
Applications
MATHEMATICS
Watch in App
Join BYJU'S Learning Program