A Nice Geometry Problem – Can You Find the Perimeter?

Hello everyone, you are welcome. Today we have a new beautiful geometry math problem. That is here we have given a right angle triangle, a closed figure whose height or perpendicular is given which is six units and the other two sides are not given. I will try to find out the perimeter of this triangle. Now as we know that the perimeter of any closed figure is the sum of its all sides, the sum of its all boundaries. So here in this figure, in this right angle triangle, we have only one side. The other two sides are not given. So here we will try to find out these two sides. But also remember that if we change this value, so there are infinite solutions for this problem. So to find out the other two sides, let us suppose this is side A, the height is equal to A. This is side B and this is side C of this triangle. So here we will try to find out the value of B and the value of C. As this is a right angle triangle, so here we can apply the Pythagoras theorem to find out the value of B and C. So let us apply the Pythagoras theorem. So by Pythagoras theorem, here we can write, so by Pythagoras theorem, Pythagoras theorem, Pythagoras theorem states that in any right angle triangle, the square of hypotenuse is equal to perpendicular square plus base square. So here our hypotenuse is C, so this will become C square is equal to perpendicular is A, so this will become A square plus base is B is equal to B square. Let's substitute the value of A, that is 6, so this will become C square is equal to the value of A is 6, so this will become 6 square plus B square. Now how do we take this B square to the left hand side? This will become C square minus B square is equal to 6 square is simply 36. Look into the left hand side here, this is C square minus B square, so this is an algebraic identity, A square minus B square. So here in the left hand side, we will use this one algebraic A square minus B square which is equal to A minus B times A plus B. So using this algebraic identity here, this left hand side will become, this will become C minus B times C plus B is equal to 36. Look into the left hand side here, there is a pair of two expressions, the product of two expressions, so therefore we will try to factorize this one number and we will find out all the possible pair of vectors of this 36. So we can write this 36 as, this is simply 1 times 36 is 36 or this can also be written as 36 times 1 is 36. Now 36 can also be written as 2 times 18 or 18 times 2 and this is also 3 times 12 which is 36 and this can also be written as 6 times 6 which is 36. So here we have four possible pair of vectors of this 36 which are these, but look into the left hand side here for any two number C and B, this subtraction, their difference is always smaller than their addition. So for that here in the right hand side we will take only those pair of numbers which suits with the left hand side. So here this pair of numbers suits this left hand side because this number is smaller than 36. This also suits, this also suits, but this one cannot suit and we cannot take this pair of number because this and this gives him the same solution. So we have three possible cases to compare the left hand side with the right hand side. One for this, second for this and third for this. But we will try to take only one pair of numbers from the right hand side. We will try to only find one solution of this problem. So let us take this one pair 2 times 18. So this equation will become, this will become C minus B times C plus B is equal to, let us take 2 times 18 which is 36. So here we will compare this expression with 2 and this expression with 18. So this will become, this gives him C minus B is equal to 2 and C plus B is equal to 18. So we will sum up these two equations for the value of B or C. So let us add these two equations. So this is the first equation. The second equation is C plus B is equal to 18. By adding these two numbers here, this negative B and this positive B will be cancelled and this will become 2C is equal to 2 plus 18 is simply 20. Dividing both sides by 2, this gives him C is equal to 20 by 2 is simply 10. So this is the value of the second variable. Now we will try to find out the value of the third unknown, that is B. So for that we will substitute this value of C in any equation. So let us substitute this value here in this one equation. So the equation is, this implies the equation is same, C plus B is equal to 18. So let us substitute C is equal to 10. This will become 10 plus B is equal to 18. Now we will take this 10 to the right hand side. So the value of B will become B is equal to, this will become 18 minus 10. 18 minus 10 is simply A. So the value of B will become A. So this is the value of B. Now here in the problem we have given the value of A and we will also find out the value of B and C from here. Now here we can very easily find out the perimeter of this closed peg or this right angled triangle. So therefore the perimeter of this right angled triangle will become, perimeter of this right angled triangle, that will become A plus B plus C. So let us substitute the value of A, B and C. So this will become here the value of A is 6, the value of B is 8 and the value of C is 10. Now summing these three numbers, 6 plus 8, it is 14. 14 plus 10, it is 24. So therefore the perimeter of this right angled triangle is, it is 24 units, which is our final answer. Now if you want to verify these values, that is, the value of A, B and C are the correct or exact values, as this is the correct answer or not. So we can also verify these values here in the Pythagoras equation. This equation C square is equal to A square plus B square. So let's try to verify these values here. So by Pythagoras theorem we can write I Fatima square is equal to perpendicular square plus base square. So here value of C is 10. So this becomes 10 square is equal to the value of A is 6, which becomes 6 square plus the value of B is 8, becomes 8 square. Now 10 square is simply 100 is equal to 6 square is 36 and 8 square is 64. So this is 100 is equal to 6 power is 10. 63 is 99 plus 1 is 10. So this is also 100. Now both sides are equal. So it's mean that the value of A, B and C are exact and correct. And last don't forget to subscribe my youtube channel for such a more interesting and step-by-step geometry solutions. Thanks for watching.