CAN YOU CALCULATE THE AREA OF THIS SQUARE?

Hi guys, let's solve one more geometry question that says the following Calculate the yellow area What we know is the following This yellow area corresponds to a square Inscribed in this triangle This side of this triangle is 13 This side is 14 And this side is 15 Then we have to find this yellow area It's a problem that has a resolution Very cool I would like you to try to solve And tell me your answer Let's see if you can Comment your answer And also comment the way you solved it Ok? Let's go to the resolution Where we start The first thing I notice Is that this triangle has the measure of the three sides So I immediately think of Heron's Theorem And that's exactly what I'm going to do What is the Heron's Theorem for? When we have a triangle And we have the three measures of the three sides And we want to find the area We can use the Heron's Theorem How does the Heron's Theorem work? First we calculate the semi-perimeter I'll call it S The Semi-perimeter What is the Semi-perimeter? It's the sum of all the measures of the sides So 13 plus 14 plus 15 This would be the perimeter 13 Plus 14 Plus 15 As it is the Heron's Theorem We need the semi-perimeter Semi is half So I'll divide this perimeter by 2 How much is this? 13 plus 14 is 27 With 15 27 plus 5 is 32 With 10 42 42 by 2 This is 21 So we have our semi-perimeter According to the Heron's Theorem What am I going to have? Let me show you The area Is the area of this triangle Is the root Of what? Of my semi-perimeter 21 times Now open the parenthesis Semi-perimeter minus One side I'll use 13 Times Semi-perimeter Minus the other side I'll use 14 Times the semi-perimeter Minus the last side In this case 15 It will be like this Right? Now just solve A is equal to the root Let me pull this 21 times 21 minus 13 8 Times 21 minus 14 7 21 minus 15 6 Perfect? Now we have to solve As I'm inside a root I'll try to Decrease these numbers By multiplying It's easier 21 7 times 3 8 2 times 2 4 times 2 8 7 6 3 times 2 Right? As I'm inside a root And everybody is multiplying Each number means That I have a square number As the root is a square I can remove this number So I have 1, 2, 7 I'll remove the 7 from the root A is equal to 7 times I have 1, 2, 3 I'll remove the 3 from the root Always removing 1 One unit I have 2 and 2 I can remove 1, 2 from the root I have 1, 2 Another 2 If I remove everybody from the root What do I do? I close the root If there was any number left I would have this multiplication Times the root of the remaining number As there was nobody left The area is 7 times 3 21 times 2 It's 42 42 times 2 It's 84 So I have the area Of my triangle If I called this triangle If I called it ABC The area of my triangle ABC Would be this 84 units of area As I defined the point ABC I'll define other points Called M, N O And P M, N, O And P As this is yellow, it's a square This part This segment M, N It's parallel to the segment B, C Right? I have here a transversal That would cut them I conclude This red angle It's also here Right? And this green angle It's also Here Right? What can I conclude? I can draw a relation I'll separate the two triangles That will be more visible The triangle A, M, N Right? And it has the green angles And the red angle I can also draw The other bigger triangle That is the whole triangle It's the triangle A B C Here It's the green angle And here It's the red angle I know that here It's 14 And we just found out that Its area Is 84 Why did I do this? With the area How can I calculate The area of a triangle? The area of a triangle So 84 Is equal to Its base 14 times Its height Would be this segment Let me draw It would be this segment Forming a 90 degree angle Right? So Its height I called it h Times h divided by 2 What do I have here? I can simplify 14 times 2 is equal to 7 84 times 7 Is equal to h 84 times 7 Is equal to 12 So h Is equal to 12 We have the value of our height So The height of this triangle Is 12 Agree? Let's go If I define the side of my square As a I defined it as a So here it will measure a Here it will measure a And here Also Let's go If the height of the big one Is 12 The height of the small one Is 12 The height of the small one Is 12 The total is 12 Which is a I'll put its height aside Its height is 12 Minus a And its base is a Here is the idea Since these two triangles are similar I'll build a relation with them and find the value of A Why do I want to find the value of A? Because A corresponds to the side of the yellow square Having the value of A I can calculate this area So, I will relate I will say the following That this part here that corresponds to the base of this triangle its base is at its height which is 12 minus A Just like now in the big triangle Its base, which is 14 is at its height, which is 12 Ok? What do I do now? Multiply, cross It will be like this, 12 times A And here I will have what? 14 times 12 minus A 12 minus A here, sorry 12 minus A What do I do? Multiply by the first and the second 12 times A is equal to 12 times 14 168 Ok Minus 14 times A 14A Ok, what do I do? I pass this minus 14A over there It will be positive So, I have 12A plus 14A This equals 168 Adding here It will be what? 26A Is equal to 168 So I will take this 26 now and pass over there dividing My A is equal to 168 divided by 26 What can I do here? I can simplify up and down by 2 Simplifying up by 2 It will be what? 84? 160 Divided by 2 is equal to 80 And here 4 is equal to 84 And down is equal to 13 So, this is the value of my side of this square 84 over 13 And now, what is the detail? We want the area So, just take the value of A How do I calculate the area? How do I calculate the area of a square? It is the value of the side In this case, it is A squared So, the area of this square It will be what? 84 over 13 squared Ok? So, A will measure what? 84 squared It will be what? 7056 And 13 squared It will be what? 169 Perfect It will be what? 7056 divided by 169 We will find an approximate value 41.75 Ok? You notice 41.75 units of area You notice half of this area This yellow area And in relation to the whole area This whole triangle It measures 84 units of area And the yellow area 41.75 That is, almost half So, great! 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