Respuesta :
Answer:
The length of shorter leg is 2.23 cm and the length of longer leg is 3.46 cm.
Step-by-step explanation:
The Pythagorean Theorem
If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.
This relationship is represented by the formula:
[tex]a^2+b^2=c^2[/tex]
Let x be length of shorter leg and 2x-1 be the length of longer leg.
By the Pythagorean Theorem,
[tex]x^2+(2x-1)^2=17[/tex]
[tex]5x^2-4x+1=17\\\\5x^2-4x-16=0[/tex]
Next, solve with the quadratic formula
[tex]\mathrm{For\:a\:quadratic\:equation\:of\:the\:form\:}ax^2+bx+c=0\mathrm{\:the\:solutions\:are\:}\\\\x_{1,\:2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}[/tex]
[tex]\mathrm{For\:}\quad a=5,\:b=-4,\:c=-16:\quad x_{1,\:2}=\frac{-\left(-4\right)\pm \sqrt{\left(-4\right)^2-4\cdot \:5\left(-16\right)}}{2\cdot \:5}[/tex]
Because the length can't be negative
[tex]x=\frac{-\left(-4\right)+\sqrt{\left(-4\right)^2-4\cdot \:5\left(-16\right)}}{2\cdot \:5}= \frac{2\left(1+\sqrt{21}\right)}{5} \approx 2.23[/tex]
The length of shorter leg is 2.23 cm and the length of longer leg is [tex]2\left(2.23\right)-1=3.46 \:cm[/tex].
The lengths of the legs of the right triangle are 8 cm and 15 cm.
Important information:
- One leg of a right triangle is 1 cm less than twice the length of the other leg.
- The hypotenuse is 17 cm long.
Pythagoras Theorem:
According to the Pythagoras Theorem:
[tex]\text{Leg}_1^2+\text{Leg}_2^2=\text{Hypotenuse}^2[/tex]
Let [tex]x[/tex] cm be the length of one leg. Then, the length of other leg is [tex]2x-1[/tex] cm.
Using the Pythagoras Theorem, we get
[tex]x^2+(2x-1)^2=(17)^2[/tex]
[tex]x^2+4x^2-4x+1=289[/tex]
[tex]5x^2-4x-288=0[/tex]
Splite the middle term.
[tex]5x^2-40x+36x-288=0[/tex]
[tex]5x(x-8)+36(x-8)=0[/tex]
[tex](5x+36)(x-8)=0[/tex]
[tex]x=-\dfrac{36}{5},8[/tex]
Side length cannot be negative. So, [tex]x=8[/tex] and the measure of other leg is:
[tex]2(8)-1=15[/tex]
Therefore, the lengths of the legs of the right triangle are 8 cm and 15 cm.
Find out more about 'Pythagoras Theorem' here:
https://brainly.com/question/15799348
