16-6 < x < 16+6
10 < x < 22 ← answer
3) SR, TR, ST(x=17, R=93, S=57, T=30, angle side inequality)
4)14,14,7 (cant be 7,7,14 because 7+7=14 so it wont for a triangle)
10 < c < 22
Remember the Triangle Inequality Theorem.
For a triangle with sides a, b, and c:
- a + b > c
- a + c > b
- b + c > a
Let's arbitrarily say a = 6 and b = 16. Then:
a + b > c
6 + 16 > c
c < 22
a + c > b
6 + c > 16
c > 10
b + c > a
16 + c > 6
c > -10
Thus, the range of possible lengths of the third side, c, is 10 < c < 22.
10 < x <22
The smallest side must be larger than the difference of the two sides
16-6 = 10
The largest side must be smaller then the sum of the two sides
Put this together
10 < x <22
10 ≥ x ≥ 22
Use the triangle inequality theorem to state the range of possibilities for the third side. If the sides of a triangle are a, b, and c, then the theorem states:
a + b ≥ c,
b + c ≥ a,
a + c ≥ b
Let's put it together then:
6 + 16 ≥ x,
so 22 ≥ x.
x + 6 ≥ 16, so
x ≥ 10.
16 + x ≥ 6, so
x ≥ -10
The 2 things in math that will never EVER be negative are times and distances, so the -10 can be disregarded.
That gives you the range of 10 ≥ x ≥ 22
That means that the third side cannot be shorter than 10 or else the triangle will not close; it also means that the third side cannot be longer than 22, or it will close but leave an "overhang".
Keep in mind that this is NOT for right triangles! To see what the third side of a right triangle is you would use Pythagorean's Theorem.
You get this by adding and subtracting the smaller side by the lengthiest side, as shown in the work below.
if you have eyes and you can use them efficiently you can see that it just looks like random dots like do a line of best fit and like they're so spread out it's such a weak association you might as well consider it nonexistent
graph them on a graphing calculator, press 2nd trace. select intersect. hit enter until it says answer. ti 84 works best