Statement
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Mathematically: AC2=AB2+BC2AC^2 = AB^2 + BC^2
Step-by-Step Derivation
1. Construction
- Consider a right triangle △ABC\triangle ABC, where ∠A=90∘\angle A = 90^\circ.
- ABAB and BCBC are the two perpendicular sides, and ACAC is the hypotenuse.
2. Algebraic Representation
The coordinates of the points are:
- A(0,0)A(0, 0): Origin.
- B(b,0)B(b, 0): On the x-axis (AB=bAB = b).
- C(0,h)C(0, h): On the y-axis (BC=hBC = h).
Using the distance formula to find the hypotenuse ACAC: AC=(b−0)2+(h−0)2AC = \sqrt{(b – 0)^2 + (h – 0)^2} AC=b2+h2AC = \sqrt{b^2 + h^2}
3. Squaring Both Sides
To eliminate the square root: AC2=b2+h2AC^2 = b^2 + h^2
Here, b2b^2 represents AB2AB^2, and h2h^2 represents BC2BC^2. Thus: AC2=AB2+BC2AC^2 = AB^2 + BC^2
Example Problem
Question
In a right triangle, the lengths of the perpendicular sides are 3 cm3 \, \text{cm} and 4 cm4 \, \text{cm}. Find the length of the hypotenuse.
Solution
Using Pythagoras Theorem: AC2=AB2+BC2AC^2 = AB^2 + BC^2
Substitute AB=3 cmAB = 3 \, \text{cm} and BC=4 cmBC = 4 \, \text{cm}: AC2=32+42AC^2 = 3^2 + 4^2 AC2=9+16=25AC^2 = 9 + 16 = 25 AC=25=5 cmAC = \sqrt{25} = 5 \, \text{cm}
Thus, the hypotenuse is 5 cm5 \, \text{cm}.
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