Right angled triangle

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

Correct result:

a =  15 cm
b =  8 cm

Solution:

$$\begin{gathered} c^2=a^2+b^2=17^2=289 \\ (c-4{)}^2=(a-3{)}^2+(b-3{)}^2=(17-4{)}^2=169 \\ 169=a^2-2\cdot 3a+3^2+b^2-2\cdot 3b+3^2 \\ 169=a^2+b^2-6\cdot a-6\cdot b+18 \\ 169-289-18=-6\cdot a-6\cdot b \\ b=-a-(-23) \\ {c}^2=289=a^2+(-a-(-23){)}^2 \\ 2a^2-46a+240=0 \\ p=2;q=-46;r=240 \\ D=q^2-4pr=46^2-4\cdot 2\cdot 240=196 \\ D>0 \\ {a}_{1,2}={\displaystyle \frac{-q\pm \sqrt{D}}{2p}}={\displaystyle \frac{46\pm \sqrt{196}}{4}} \\ {a}_{1,2}={\displaystyle \frac{46\pm 14}{4}} \\ {a}_{1,2}=11.5\pm 3.5 \\ {a}_1=15 \\ {a}_2=8 \\ \text{ Factored form of the equation: } \\ 2(a-15)(a-8)=0 \\ a=a_1=15\text{ cm } \\ b=a_2=8cm \end{gathered}$$

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