One side of a triangle is 1 cm shorter than the base, x . The other side is 9 longer than the base. What lengths of the base will allow the perimeter of the triangle to be at least 20 cm?

Question

One side of a triangle is 1 cm shorter than the base, xx. The other side is 9 longer than the base. What lengths of the base will allow the perimeter of the triangle to be at least 20 cm?

I have to know this for math but I don't understand how to do it and I really need help ASAP.

Answer 1

Draw a triangle. Label the bottom base "x", label the left side "x-1" and label the right side "x+9".

x+(x-1)+(x+9)>=20 which becomes 3x+8>=20, 3x>=12, x>=4

But x>=4 is not the answer because there is another condition that has to be met.

The sum of the measures of any two sides of a triangle must be greater than the measure of the third side.

The longest side is "x+9".

Therefore x+(x-1)>x+9 is the second condition that has to be met.

So, 2x-1>x+9, 2x>x+10, x>10

Therefore both conditions x>=4 and x>10 must be met.

Therefore the solution is x>10 also satisfies x>=4 because any number greater than 10 is also greater than 4.

Check by choosing a number greater than 10, such as 11.

The base is 11cm, the left side is 10cm and the right side is 20 cm and the perimeter is 11+10+20=41cm.

Any number 10 or less will not satisfy the second condition.

Answer 2

One side of a triangle is 1 cm shorter than the base, xx. The other side is 9 longer than the base. What lengths of the base will allow the perimeter of the triangle to be at least 20 cm?

Answer:

So the base length of the base is x. Another side is x+9 and the other side is x-1. So if we solve for a 20cm perimeter, we add up all the sides to be greater or equal to 20.

x+x-1+x+9 >= 20

3x + 8 >= 20

3x + 8 - 8 >= 20 - 8

3x >= 12

x >= 4 cm

Answer 3

a=x-1= one side

b= x+9= another side

x = the base

Perimeter = sum of sides

=a+b+x

= x-1+x+9+x

= 3x +8 =20

3x = 20-8 = 12

x = 12/3 = 4 cm

the base has to be at least 4 cm long

but that means the sides are 4, 3 and 13 which is impossible

as no side can be smaller than the sum of the other 2 sides

13>4+3=7

13>7

x = x+9 + x-1

x= 2x-8

x = 8 cm

x has to be greater than or equal to 8 cm

x > 8 cm

but that won't work either, as then the sides are 8,7 and 17

were 17>8+7

x has to be greater than 10

x> 10

if x= 10, then the other sides are 9 and 19

but that's a degenerate "triaangle" that collapses into a line

so the base has to be larger than 10 cm

this is for algebra I? Must be extra credit?

that's not a normal problem in algebra 1

unless common core is going crazy again

or your instructor is a tad sadistic