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प्रश्न
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Use of mobile screen for long hours makes your eyesight weak and gives you headaches. Children who are addicted to playing ‘BGMI’ can get easily stressed out. To raise social awareness about ill effects of playing BGMI, a school decided to start ‘BAN BGMI’ campaign, in which students are asked to prepare campaign board in the shape of a rectangle. One such campaign board made by class X students of the school is shown in the figure given below.
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Based on the above information, answer the questions given below.
- What are the coordinates of the point of intersection of diagonals AC and BD?
- What is the length of diagonal AC?
- Find the ratio AB : AD.
- Find the area of the campaign board ABCD.
- What is the perimeter of the campaign board ABCD?
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उत्तर
From the given graph, the coordinates of the vertices of rectangle ABCD are: \[A(1, 1), \quad B(7, 1), \quad C(7, 5), \quad D(1, 5)\]
i. Point of intersection of diagonals AC and BD:
In a rectangle, diagonals bisect each other, so their intersection point is the midpoint of diagonal AC.
Midpoint Formula: \[\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) = \left(\frac{1 + 7}{2}, \frac{1 + 5}{2}\right)\]
\[= \left(\frac{8}{2}, \frac{6}{2}\right)\]
\[= (4, 3)\]
ii. Length of diagonal (AC):
Distance Formula: \[AC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
\[AC = \sqrt{(7 - 1)^2 + (5 - 1)^2}\]
\[AC = \sqrt{6^2 + 4^2}\]
\[= \sqrt{36 + 16}\]
\[= \sqrt{52}\]
\[= 2\sqrt{13}\text{ units}\]
iii. Ratio (AB : AD):
Length of side (AB): \[AB = \sqrt{(7 - 1)^2 + (1 - 1)^2}\]
\[= \sqrt{6^2}\]
\[= 6\text{ units}\]
Length of side (AD): \[AD = \sqrt{(1 - 1)^2 + (5 - 1)^2}\]
\[= \sqrt{4^2}\]
\[= 4\text{ units}\]
Ratio: \[AB : AD = 6 : 4\]
\[= 3 : 2\]
iv. Area of the campaign board ABCD:
Formula: \[\text{Area} = \text{Length} \times \text{Breadth} = AB \times AD\]
Calculation: \[\text{Area} = 6 \times 4 = 24\text{ sq units}\]
v. Perimeter of the campaign board ABCD:
Formula: \[\text{Perimeter} = 2(AB + AD)\]
Calculation: \[\text{Perimeter} = 2(6 + 4)\]
\[= 2(10)\]
\[= 20\text{ units}\]
