The lengths of the three sides of a triangle are in the ratio 3:4:x, where x is an integer… — ACT

The lengths of the three sides of a triangle are in the ratio $3:4:x$ , where $x$ is an integer. If the perimeter of the triangle is $36$ centimeters, how many different possible values can $x$ have?

A
1
B
3
2Answer
D
4
E
5

Answer

There are 2 possible integer values for

x

The correct answer is the value of 2. By expressing the side lengths as

3k

4k

, and

xk

where

k

must be a positive integer, the perimeter equation becomes

k(7 + x) = 36

. Solving for

7 + x

yields factors of

36

greater than

7

, which are

9, 12, 18,

and

36

. These correspond to

x

values of

2, 5, 11,

and

29

. Checking each set of side lengths against the Triangle Inequality Theorem shows that only the sets corresponding to

x = 2

(sides

12, 16, 8

) and

x = 5

(sides

9, 12, 15

) form valid triangles.

Step-by-Step Solution

Define the side lengths using a multiplier

k

Let the side lengths of the triangle be

3k

4k

, and

xk

for some positive multiplier

k

. Since all three side lengths must be integers, the difference between the first two sides,

4k - 3k = k

, must also be an integer. Thus,

k

must be a positive integer.

This establishes that the scaling factor

k

is a positive integer, allowing us to find discrete solutions.

Set up the perimeter equation and express

x

in terms of

k

The perimeter is the sum of the side lengths:

3k + 4k + xk = 36 \implies k(7 + x) = 36

. Since

k

and

x

are positive integers,

7 + x

must be a factor of

36

that is greater than

7

This constrains the possible values of

x

to the factors of

36

that are larger than

7

Find the potential values of

x

and their corresponding side lengths.

The factors of

36

greater than

7

are

9, 12, 18,

and

36

. This yields four potential cases:
- If

7+x = 9 \implies x = 2

, then

k = 4

, and the sides are

12, 16, 8

.
- If

7+x = 12 \implies x = 5

, then

k = 3

, and the sides are

9, 12, 15

.
- If

7+x = 18 \implies x = 11

, then

k = 2

, and the sides are

6, 8, 22

.
- If

7+x = 36 \implies x = 29

, then

k = 1

, and the sides are

3, 4, 29

This identifies all mathematically possible configurations before checking if they can physically form a triangle.

Apply the Triangle Inequality Theorem to each case.

The sum of the lengths of any two sides must be strictly greater than the length of the remaining side:
- For sides

12, 16, 8

12 + 8 = 20 > 16

(Valid).
- For sides

9, 12, 15

9 + 12 = 21 > 15

(Valid).
- For sides

6, 8, 22

6 + 8 = 14 < 22

(Invalid).
- For sides

3, 4, 29

3 + 4 = 7 < 29

(Invalid).
Only the cases where

x = 2

and

x = 5

form valid triangles.

This filters the candidate values of

x

to only those that can form a valid geometric triangle.

Key Concept

Triangle Inequality Theorem and Integer Ratio Constraints

Estimated Time:1m 30s

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