Question
Download Solution PDFज्याचे क्षेत्रफळ 24 सेमी2 आहे आणि त्याच्या कर्णांच्या लांबीची बेरीज 14 सेमी आहे अशा समभुज चौकोनाच्या एका बाजूची लांबी शोधा.
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFदिलेल्याप्रमाणे:
समभुज चौकोनाचे क्षेत्रफळ = 24 सेमी2
त्याच्या कर्णांच्या लांबीची बेरीज = 14 सेमी
वापरलेली संकल्पना:
1. समभुज चौकोनाचे कर्ण नेहमी 90° च्या कोनात एकमेकांना दुभागतात.
2. समभुज चौकोनाचे क्षेत्रफळ = \(\frac {1}{2} \times\) त्याच्या कर्णांचा गुणाकार
3. (a + b)2 = (a - b)2 + 4ab
गणना:
समभुज चौकोनाच्या कर्णांची लांबी अनुक्रमे P आणि Q मानू (जेथे P > Q)
संकल्पनेनुसार,
PQ/2 = 24
⇒ PQ = 48
प्रश्नानुसार,
P + Q = 14 ---(1)
⇒ (P + Q)2 = 196
⇒ (P - Q)2 + 4PQ = 196
⇒ (P - Q)2 = 196 - 4PQ
समीकरणामध्ये, PQ चे मूल्य ठेवू,
⇒ (P - Q)2 = 196 - 4 × 48
⇒ (P - Q)2 = 196 - 192
⇒ (P - Q)2 = 4
⇒ (P - Q) = ± 2
⇒ (P - Q) = 2 [∵ P > Q, +2 घेऊ] ---(2)
समीकरण (1) आणि (2) ची बेरीज केल्यास, आपल्याकडे,
P + Q + P - Q = 14 + 2
⇒ 2P = 16
⇒ P = 8 सेमी
अशाप्रकारे, Q = 8 - 2 = 6 सेमी
संकल्पनेनुसार,
ΔCOD हा बिंदु O वर एक काटकोन त्रिकोण आहे.
अशाप्रकारे, DO = OB = 8/2 = 4 सेमी
AO = OC = 6/2 = 3 सेमी
आता, DC = \(\sqrt {4^2 + 3^2}\) = 5 सेमी
∴ समभुज चौकोनाच्या प्रत्येक बाजूची लांबी 5 सेमी आहे.
Last updated on Jul 21, 2025
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