Right angled triangle

The hypotenuse of a right triangle is 17 cm long. When we decrease the length of the legs by 3 cm, then decrease its hypotenuse by 4 cm. Find the size of its legs.

Final Answer:

a =  15 cm
b =  8 cm

Step-by-step explanation:

$$\begin{gathered} c^2=a^2+b^2=17^2=289 \\ (c-4{)}^2=(a-3{)}^2+(b-3{)}^2=(17-4{)}^2=169 \\ 169=a^2-2\cdot 3a+3^2+b^2-2\cdot 3b+3^2 \\ 169=a^2+b^2-6\cdot a-6\cdot b+18 \\ 169-289-18=-6\cdot a-6\cdot b \\ b=-a-(-23) \\ {c}^2=289=a^2+(-a-(-23){)}^2 \\ 2a^2-46a+240=0 \\ 2...\text{ prime number} \\ 46=2\cdot 23 \\ 240=2^4\cdot 3\cdot 5 \\ \text{GCD}(2,46,240)=2=2 \\ {a}^2-23a+120=0 \\ p=1;q=-23;r=120 \\ D=q^2-4pr=23^2-4\cdot 1\cdot 120=49 \\ D>0 \\ {a}_{1,2}=\frac{-q\pm \sqrt{D}}{2p}=\frac{23\pm \sqrt{49}}{2} \\ {a}_{1,2}=\frac{23\pm 7}{2} \\ {a}_{1,2}=11.5\pm 3.5 \\ {a}_1=15 \\ {a}_2=8 \\ \text{ Factored form of the equation: } \\ (a-15)(a-8)=0 \\ a=a_1=15\text{cm} \\ b=a_2=8cm \end{gathered}$$

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