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Mathematics · 2024 · 3 marks
CBSE 2024 · Region 1 · Set 1 · Q26
(A) Find the ratio in which the point (8/5, y) divides the line segment joining the points (1,2) and (2,3). Also, find the value of y.
(A)
Find the ratio in which the point $\displaystyle \left(\frac{8}{5}, y\right)$ divides the line segment joining the points $\displaystyle (1,2)$ and $\displaystyle (2,3)$. Also, find the value of $\displaystyle y$.
OR
(B)
ABCD is a rectangle formed by the points $\displaystyle \mathrm{A}(-1,-1), \mathrm{B}(-1,6)$, C $\displaystyle (3,6)$ and D $\displaystyle (3, -1)$. P, Q, R and S are mid-points of sides AB, BC, CD and DA respectively. Show that diagonals of the quadrilateral PQRS bisect each other.
Marking-scheme solution
Let AP: \(\displaystyle \mathrm{PB}=\mathrm{k}: 1\)
\[\begin{aligned}
& \therefore \frac{2 k+1}{k+1}=\frac{8}{5} \\
& \Rightarrow \mathrm{k}=\frac{3}{2}
\end{aligned}
\]
∴ required ratio is $\displaystyle 3$: 2.
\[y=\frac{3 \times 3+2 \times 2}{3+2}=\frac{13}{5}
\]
Co-ordinates of point P are \(\displaystyle \left(\frac{-1-1}{2}, \frac{-1+6}{2}\right)\) i.e. \(\displaystyle \left(-1, \frac{5}{2}\right)\)
Co-ordinates of point Q are \(\displaystyle \left(\frac{-1+3}{2}, \frac{6+6}{2}\right)\) i.e. \(\displaystyle (1,6)\)
Co-ordinates of point R are \(\displaystyle \left(\frac{3+3}{2}, \frac{6-1}{2}\right)\) i.e. \(\displaystyle \left(3, \frac{5}{2}\right)\)
Co-ordinates of point S are \(\displaystyle \left(\frac{-1+3}{2}, \frac{-1-1}{2}\right)\) i.e. \(\displaystyle (1,-1)\)
Co-ordinates of mid point of diagonal QS are \(\displaystyle \left(\frac{1+1}{2}, \frac{6-1}{2}\right)\) i.e. \(\displaystyle \left(1, \frac{5}{2}\right)\)
Co-ordinates of mid point of diagonal PR are \(\displaystyle \left(\frac{-1+3}{2}, \frac{\dfrac{5}{2}+\dfrac{5}{2}}{2}\right)\) i.e. \(\displaystyle \left(1, \frac{5}{2}\right)\)
Since coordinates of mid point of QS = coordinates of mid point of PR Therefore, diagonals PR and QS bisect each other.
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