{"id":2429,"date":"2024-03-05T15:32:27","date_gmt":"2024-03-05T15:32:27","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=2429"},"modified":"2024-03-05T15:32:27","modified_gmt":"2024-03-05T15:32:27","slug":"a-point-is-selected-at-random-inside-a-rectangle-and-perpendiculars-are-drawn-on-each-side-of-the-rectangle-from-the-point-the-total-length-of-these-perpendiculars-is-40-cm-if-one-side-of-the-rectan","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/a-point-is-selected-at-random-inside-a-rectangle-and-perpendiculars-are-drawn-on-each-side-of-the-rectangle-from-the-point-the-total-length-of-these-perpendiculars-is-40-cm-if-one-side-of-the-rectan\/","title":{"rendered":"A point is selected at random inside a rectangle and perpendiculars are drawn on each side of the rectangle from the point. The total length of these perpendiculars is 40 cm. If one side of the rectangle is3 times the other, what will be the area of the rectangle ?"},"content":{"rendered":"<p>\r\n    <!-- Check if it's an AMP page -->\r\n            <!-- Non-AMP version -->\r\n        <div class=\"mcq-container\" data-quiz-id=\"quizState_6a985b2b9edbb\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"false\">\r\n                    300 sq. cm.                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"false\">\r\n                    350 sq. cm.                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option3\" data-is-correct=\"false\">\r\n                    192 sq. cm.                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option4\" data-is-correct=\"true\">\r\n                    None of the above                <\/div>\r\n                            \r\n            <!-- Feedback messages for non-AMP -->\r\n            <div class=\"feedback\" data-feedback=\"wrong\">Answer is Right!<\/div>\r\n            <div class=\"feedback\" data-feedback=\"right\">Answer is Wrong!<\/div>\r\n        <\/div>\r\n\r\n        <script>\r\n        document.addEventListener('DOMContentLoaded', function () {\r\n            var containers = document.querySelectorAll('.mcq-container');\r\n\r\n            containers.forEach(function(container) {\r\n                var options = container.querySelectorAll('.option');\r\n                var feedbackSelect = container.querySelector('[data-feedback=\"select\"]');\r\n                var feedbackWrong = container.querySelector('[data-feedback=\"wrong\"]');\r\n                var feedbackRight = container.querySelector('[data-feedback=\"right\"]');\r\n\r\n                options.forEach(function(option) {\r\n                    option.addEventListener('click', function() {\r\n                        var selectedOption = option.getAttribute('data-option-key');\r\n                        var isCorrect = option.getAttribute('data-is-correct') === 'true';\r\n\r\n                        \/\/ Remove previous selections\r\n                        options.forEach(function(opt) {\r\n                            opt.classList.remove('correct', 'incorrect');\r\n                        });\r\n\r\n                        \/\/ Add the correct\/incorrect class\r\n                        if (isCorrect) {\r\n                            option.classList.add('correct');\r\n                            feedbackRight.hidden = false;\r\n                            feedbackWrong.hidden = true;\r\n                        } else {\r\n                            option.classList.add('incorrect');\r\n                            feedbackRight.hidden = true;\r\n                            feedbackWrong.hidden = false;\r\n                        }\r\n\r\n                        \/\/ Hide select feedback\r\n                        feedbackSelect.hidden = true;\r\n                    });\r\n                });\r\n            });\r\n        });\r\n        <\/script>\r\n    \r\n    <!--more--><\/p>\n<p>The correct answer is (d).<\/p>\n<p>Let $l$ be the length of the shorter side of the rectangle and $3l$ be the length of the longer side. The area of the rectangle is $l(3l) = 3l^2$.<\/p>\n<p>The perpendiculars drawn from the point inside the rectangle to the sides form four right triangles. The hypotenuses of these triangles are all equal to the length of the diagonal of the rectangle, which is $\\sqrt{l^2 + (3l)^2} = 2\\sqrt{10}l$.<\/p>\n<p>The sum of the lengths of the perpendiculars is $40$ cm. This means that the sum of the lengths of the legs of the four right triangles is $40$ cm.<\/p>\n<p>The legs of a right triangle can be any two numbers that add up <div class=\"youtube-subscribe-container\">\r\n        <a href=\"https:\/\/www.youtube.com\/channel\/UCNHT8lW-JmLC68rjBfZhdkg?sub_confirmation=1\" target=\"_blank\" class=\"youtube-subscribe-button\">\r\n            <span class=\"youtube-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 576 512\">\r\n                   <div class=\"telegram-channel-container\">\r\n        <a href=\"https:\/\/t.me\/pscnotes2025\" target=\"_blank\" class=\"telegram-channel-button\">\r\n            <span class=\"telegram-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 496 512\">\r\n                    <path fill=\"white\" d=\"M248,8C111,8,0,119,0,256s111,248,248,248s248-111,248-248S385,8,248,8z M362,177L320,367c-3,14-10,18-20,14l-56-41l-27,26 c-3,3-5,5-10,5l4-63L323,196c5-5-1-7-8-3l-98,62l-42-13c-9-3-10-9,2-14l162-63C351,160,365,164,362,177z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Join Our Telegram Channel\r\n        <\/a>\r\n    <\/div>  <path d=\"M549.7 124.1c-6.3-23.7-24.8-42.3-48.3-48.6C458.8 64 288 64 288 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Subscribe on YouTube\r\n        <\/a>\r\n    <\/div> to $2\\sqrt{10}l$. There are infinitely many such pairs of numbers, so there are infinitely many possible rectangles that satisfy the given conditions.<\/p>\n<p>The area of any of these rectangles is $3l^2$. Therefore, the area of the rectangle could be any number greater than or equal to $0$.<\/p>\n<p>The options (a), (b), and (c) are all specific values for the area of the rectangle. However, as we have seen, the area of the rectangle could be any number greater than or equal to $0$, so none of these options is correct.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Join Our Telegram Channel Subscribe on YouTube<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[41],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>A point is selected at random inside a rectangle and perpendiculars are drawn on each side of the rectangle from the point. The total length of these perpendiculars is 40 cm. 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