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 these legs.

Final Answer:

a =  15 cm
b =  8 cm

Step-by-step explanation:

$$\begin{gathered} c=17\text{cm} \\ {c}^2=a^2+b^2 \\ (17-4{)}^2=(a-3{)}^2+(b-3{)}^2 \\ 289=a^2+b^2 \\ 169=a^2-6a+9+b^2-6b+9 \\ 289=a^2+b^2 \\ 151=a^2+b^2-6a-6b \\ 151=289-6a-6b \\ 138=6a+6b \\ 23=a+b \\ 289=a^2+b^2 \\ 289=a^2+(23-a{)}^2 \\ 289=a^2+23^2-46a+a^2 \\ 2a^2-46a+240=0 \\ 2a^2-46a+240=0 \\ 2...\text{ prime number} \\ 46=2\cdot 23 \\ 240=2^4\cdot 3\cdot 5 \\ \text{GCD}(2,46,240)=2 \\ {a}^2-23a+120=0 \\ p=1;q=-23;r=120 \\ D=q^2-4pr=23^2-4\cdot 1\cdot 120=49 \\ D>0 \\ {a}_{1,2}=\frac{-q\pm \sqrt{D}}{2p}=\frac{23\pm \sqrt{49}}{2} \\ {a}_{1,2}=\frac{23\pm 7}{2} \\ {a}_{1,2}=11.5\pm 3.5 \\ {a}_1=15 \\ {a}_2=8 \\ a=a_1=15=15\text{cm} \end{gathered}$$

Our quadratic equation calculator calculates it.

$b=\sqrt{c^2-a^2}=\sqrt{17^2-15^2}=8\text{cm}$

Try calculation via our triangle calculator.

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