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KCC's Quizzes AQQ290 about a triangle coming from the Hell

		
	
				

			

									
					    KCC
					
							

            
            			
				
									
											
			
			
				
  •  Analog Employees  

			
			
			    	
			                    
                
				                on Aug 22, 2025 
				
			
			
		

		
		                     
				
        		

		    
		    		    
		    			

			    
So, after a long (and deserved) summer vacation, we can suppose your brain is now fully energized and ready to tackle complex problem! 


OK, I know the 2 famous laws about vacations!


1. Vacations are always too short (even if you had 4 weeks)!


2. They always occur in wrong moments (i.e. you were disturbed by any kind of urgencies).


1. First, the quote of the week: "People often say that motivation doesn't last. Well, neither does bathing - that's why we recommend it daily" - Zig Ziglar


                                                         


                                             Source: encrypted-tbn0.gstatic.com/images


2. New challenge: it's a geometry problem, about a triangle that comes from hell... Be careful, there is a trap...


                        


A right triangle has a hypothenuse of 10 cm and an altitude to the hypothenuse of 6 cm.


Question: What is the area of the triangle ABC?




Good luck! And try to be among the first ones!


Also, don't hesitate to share such quizzes with colleagues or friends!

			

			
									
		                             
                        

    

                                            
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    •  Analog Employees  
    
            
						
                            
            
            				
            
                on Aug 24, 2025 5:29 AM
            
            
						
			            		
    
		
								
	
    

		
		
    
		
    
                                
                                                            
    
					
    The triangle cannot exist as a hypothenuse of 10cm is not compatible with an altitude of 6cm. In fact, say x= cathetus 1, y= cathetus 2, t= hypothenuse, h= altitude to the hypothenuse, A= area of triangle ABC. Then you have A=x*y/2 and A=t*h/2. Also, from Pythagoras you have: x^2 + y^2 = t^2. By using this data, you end up in the equation: x^4 - t^2 * x^2 + (t*h)^2 =0. Replace z=x^2 and then you have: z^2 - t^2 * z + (t*h)^2 = 0. Now, this equation doesn't have any real solution, its discriminant being Δ=sqrt (t^4 - 4* t^2 * h^2) < 0. Since t=10cm, you can see that the maximum value of h for the existence of the triangle is any value lower than 5cm.  
    
			
    

		
			
	      

		
		
	
    
	
    
		
    
			
		
    
	
    
	
    
		
						
				
				
						
				
				
		
		
    
			
		
    
	
    

    

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    •  Analog Employees  
    
            
						
                            
            
            				
            
                on Aug 24, 2025 5:29 AM
            
            
						
			            		
    
		
								
	
    

		
		
    
		
    
                                
                                                            
    
					
    The triangle cannot exist as a hypothenuse of 10cm is not compatible with an altitude of 6cm. In fact, say x= cathetus 1, y= cathetus 2, t= hypothenuse, h= altitude to the hypothenuse, A= area of triangle ABC. Then you have A=x*y/2 and A=t*h/2. Also, from Pythagoras you have: x^2 + y^2 = t^2. By using this data, you end up in the equation: x^4 - t^2 * x^2 + (t*h)^2 =0. Replace z=x^2 and then you have: z^2 - t^2 * z + (t*h)^2 = 0. Now, this equation doesn't have any real solution, its discriminant being Δ=sqrt (t^4 - 4* t^2 * h^2) < 0. Since t=10cm, you can see that the maximum value of h for the existence of the triangle is any value lower than 5cm.  
    
			
    

		
			
	      

		
		
	
    
	
    
		
    
			
		
    
	
    
	
    
		
						
				
				
						
				
				
		
		
    
			
		
    
	
    

    

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