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Mathematics · 2023 · 4 marks
CBSE 2023 · Region 4 · Set 1 · Q37
Jagdish has a field which is in the shape of a right angled triangle AQC. He wants to leave a space in the form of a square PQRS inside the field for…
Jagdish has a field which is in the shape of a right angled triangle AQC. He wants to leave a space in the form of a square PQRS inside the field for growing wheat and the remaining for growing vegetables (as shown in the figure). In the field, there is a pole marked as O.
Based on the above information, answer the following questions :
(i)
Taking O as origin, coordinates of P are $\displaystyle (-200, 0)$ and of Q are $\displaystyle (200, 0)$. PQRS being a square, what are the coordinates of R and S ?
(ii)
(a)
What is the area of square PQRS ?
OR
(b)
What is the length of diagonal PR in square PQRS ?
(iii)
If S divides CA in the ratio K:$\displaystyle 1$, what is the value of K, where point $\displaystyle A$ is $\displaystyle (200,800)$ ?
Marking-scheme solution
(i) \(\displaystyle \mathrm{R}(200,400), \mathrm{S}(-200,400)\)
(ii) (a) side \(\displaystyle \mathrm{PQ}=(200+200) \mathrm{m}=400 \mathrm{~m}\)
Area of square \(\displaystyle \mathrm{PQRS}=400 \times 400\)
\[\text { = } 160000 \text { sq. units }
\]
(ii) (b)
\[\begin{aligned}
\text { Diagonal PR } & =\sqrt{(400)^{2}+(400)^{2}} \\
& =\sqrt{3200} \text { or } 400 \sqrt{2}
\end{aligned}
\]
(iii) \(\displaystyle C(-600,0) ; A(200,800) ; S(-200,400)\)
S divides CA in the ratio \(\displaystyle k: 1\)
\[\begin{aligned}
-200 & =\frac{k(200)+1(-600)}{k+1} \\
\Rightarrow k & =1
\end{aligned}
\]
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