RTriangle 17

The hypotenuse of a right triangle is 17 cm. If you decrease both two legs by 3 cm you will reduce the hypotenuse by 4 cm. Determine the length of this legs.

Correct answer:

a =  15 cm
b =  8 cm

Step-by-step explanation:

$$\begin{gathered} 17^2=a^2+b^2 \\ (17-4{)}^2=(a-3{)}^2+(b-3{)}^2 \\ 289=a^2+b^2 \\ 169=a^2-6a+9+b^2-6b+9 \\ 289=a^2+b^2 \\ 151=a^2+b^2-6a-6b \\ 151=289-6a-6b \\ 138=6a+6b \\ 23=a+b \\ 289=a^2+b^2 \\ 289=a^2+(23-a{)}^2 \\ 289=a^2+23^2-46a+a^2 \\ 2a^2-46a+240=0 \\ 2a^2-46a+240=0 \\ p=2;q=-46;r=240 \\ D=q^2-4pr=46^2-4\cdot 2\cdot 240=196 \\ D>0 \\ {a}_{1,2}={\displaystyle \frac{-q\pm \sqrt{D}}{2p}}={\displaystyle \frac{46\pm \sqrt{196}}{4}} \\ {a}_{1,2}={\displaystyle \frac{46\pm 14}{4}} \\ {a}_{1,2}=11.5\pm 3.5 \\ {a}_1=15 \\ {a}_2=8 \\ \text{ Factored form of the equation: } \\ 2(a-15)(a-8)=0 \\ a=15cm \\ b=8cm \end{gathered}$$

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