Question

Please help if you only answer one is still very appreciated♥️​​

Answer 1

1)
`9 < x < 31`

2)
`10 < x < 54`

3) Not a triangle

4) This is a triangle

5)
`6√(2)`

6)
`4√(6)`

7)
`x=√(296) \\x=2√(74) \\x=17.2047`

8)
`x=7`

9) Obtuse

10) Obtuse

Explanation:

1) This one is a bit difficult to explain without a picture, but the third side needs to be longer than the difference of the 2 sides, and shorter than the sum of the 2 sides.

`20+11=31\\20-11=9`

The third side must be shorter than 31 and longer than 9.

`9 < x < 31`

2) Same thing here.

`32+22=54\\32-22=10\\10 <  x < 54`

3) Use the same method as the first 2 questions.

`10 + 5 = 15\\10 - 5 = 5`

15 is greater than 5, but not less than 15. Not a triangle.

4) Same thing again.

`12+2.6=14.6\\12-2.6=9.4`

12 is greater than 9.4 and less than 14.6. This is a triangle.

5) Split 72 into its prime factors, rewrite any pairs in exponent form, and apply
`√(x^2) =x`

`√(72) \\√(2*2*2*3*3)\\√((2*2)*(3*3)*2) \\√(2^2*3^2*2) \\(2*3)√(2) \\6√(2)`

6) Same thing once again.

`√(96)\\√(2*2*2*2*2*3)\\√((2*2)*(2*2)*2*3)\\√(2^2*2^2*2*3) \\(2*2)√(2*3)  \\4√(6)`

7) Use the pythagorean theorem (
`a^2+b^2=c^2`)

`14^2+10^2=x^2\\196+100=x^2\\x^2=296\\x=√(296) ,-√(296)`

Distances cant be negative, so
`x=√(296) \\x=2√(74)`

8) The time to repeat the same method has come.

`5^2+x^2=√(74) ^2\\x^2+25=74\\x^2=49\\x=7,-7`

Distances cant be negative, so
`x=7`

9) Check distances with the pythagorean theorem.

`12^2+9^2=17^2\\144+81=289\\225\\eq289`

The third side is too long to be a right triangle, so this triangle is obtuse.

10) Repeat the previous method once more.

`6^2+(2√(55) )^2=17^2\\36+220=289\\256\\eq 289`

Third side is too long, triangle is obtuse.

Answer 2

3. NO

4. NO

7. c = 17.2

8. x = 7

Explanation: