Pythagorean Theorem and Quadratic Equation Introduction Discovered and developed by scientist and mathematician, Pythagoras (c. 570 BC – c. 495 BC),‘Pythagorean Theorem’ is a widely applied mathematical statement which illustrates the relation of the three sides of a right triangle that consists of two legs and a longest side, known as the hypotenuse. In equation, it is given by —
c2 = a2 + b2
where the variables ‘a’ and ‘b’ refer to the length measures of the right triangle’s legs while the variable ‘c’ pertains to the hypotenuse. Applications of Pythagorean Theorem are recognized in various fields of maths such as algebra, trigonometry, and calculus.
Problem 98: Buried Treasure
Ahmed has half of a treasure map, which indicates that the treasure is buried in the desert 2x + 6 paces from Castle Rock. Vanessa has the other half of the map. Her half indicates that to find the treasure, one must get to Castle Rock, walk x paces to the north, and then walk 2x + 4 paces to the east. If they share their information, then they can find x and save a lot of digging. What is ‘x’?
Solution:
By Pythagorean Theorem, x2 + (2x + 4)2 = (2x + 6)2
— x2 + 4×2 + 16x + 16 = 4×2 + 24x + 36
Combining similar terms,
— x2 + 16x – 24x + 16 – 36 = 0
Reduces to Quadratic Equation: x2 – 8x – 20 = 0
where the left side can be factored into (x – 10) * (x + 2) = 0
Then by Zero-Factor Rule: x – 10 = 0 and x + 2 = 0
Isolating ‘x’ in each equation gives x = 10 and x = -2
Taking the positive value, x = 10 paces
Since the Pythagorean Theorem is given by c2 = a2 + b2 then, each of the expressions ‘x’, ‘2x + 4’, and ‘2x + 6’ representing the sides of the right triangle which encloses the route to the buried treasure may be plugged into the Pythagorean equation such that (2x + 6)2 = x2 + (2x + 4)2 where ‘2x + 6’ paces refers to the measure of the longest side. Then expanding the binomials (2x + 6)2 and (2x + 4)2 yields 4×2 + 24x + 36 and 4×2 + 16x + 16, respectively. Upon combining like terms in the compound equation formed, the resulting equation turns out quadratic with x2 – 8x – 20 = 0.
By factoring the trinomial, x2 – 8x – 20 becomes the product (x – 10) * (x + 2). Through zero-factor property, each factor may be equated to zero to have x – 10 = 0 and x + 2 = 0, correspondingly. Solving completely, ‘x’ can be isolated on one side of each equation, becoming x = 10 and x = -2. It is logical to use positive values, so in this case, take x = 10. This means from Castle Rock to the place where the treasure is buried, Ahmed can walk 2*(10) + 6 or 26 paces to access the treasure or Vanessa can walk 10 paces heading north first then 2*(10) + 4 or 24 paces going east to be brought to the treasure spot.
Conclusion
Apparently, ‘Pythagorean Theorem’ proves useful in solving the specified problem which may be put into an illustration of a closed three-sided figure. Since there are distances covered northward and eastward, a 90-degree angle forms out of these segments, so that a right triangle is eventually created once the hypotenuse is drawn to connect the initial point (origin) and the final point (end). Such may be used in comparing the practicality of taking a straight route (along the hypotenuse) to the impracticality of taking two paths (shorter legs) which would consume more time of travel.