A non-degenerate triangle has side lengths of 5, 12, and x. A second non-degenerate triang… — ACT

A non-degenerate triangle has side lengths of $5$ , $12$ , and $x$ . A second non-degenerate triangle has side lengths of $x$ , $10$ , and $y$ . If $x$ and $y$ must be integers, and the perimeter of the second triangle is the minimum possible integer value, what is the sum of all possible values of $y$ ?

A
1
B
3
6Answer
D
10
E
27

Answer

The correct answer is 6. By applying the Triangle Inequality Theorem, the shared side length

x

of the first triangle must satisfy

12 - 5 < x < 12 + 5

, which simplifies to

7 < x < 17

. Since

x

must be an integer, its possible values are

\{8, 9, 10, 11, 12, 13, 14, 15, 16\}

. For the second triangle with side lengths

x

10

, and

y

, the Triangle Inequality Theorem requires

|x - 10| < y < x + 10

. To minimize the perimeter

P = x + 10 + y

, we minimize

x + y

. Checking the possible values of

x

, we find that when

x = 8

, the minimum integer value for

y

3

(giving

P = 21

); when

x = 9

, the minimum integer value for

y

2

(giving

P = 21

); and when

x = 10

, the minimum integer value for

y

1

(giving

P = 21

). For any

x \ge 11

, the minimum value of

y

x - 9

, resulting in a perimeter of at least

23

. Therefore, the minimum perimeter of the second triangle is

21

, which is achieved when

y

3

2

, or

1

. The sum of these values of

y

1 + 2 + 3 = 6

Step-by-Step Solution

Apply the Triangle Inequality Theorem to the first triangle to find the range of the shared side length

x

7 < x < 17

The sum of any two side lengths of a triangle must be strictly greater than the third side length. Thus,

12 - 5 < x < 12 + 5

List the possible integer values of

x

based on the range.

x \in \{8, 9, 10, 11, 12, 13, 14, 15, 16\}

The problem states that

x

must be an integer.

Apply the Triangle Inequality Theorem to the second triangle with sides

x

10

, and

y

to express the range of

y

in terms of

x

|x - 10| < y < x + 10

The third side length

y

must be strictly between the difference and the sum of the other two sides (

x

and

10

Determine the minimum integer value of the perimeter of the second triangle,

P = x + 10 + y

, by testing the possible values of

x

and finding the minimum integer

y

for each.

The minimum perimeter is

21

, achieved when

(x, y) = (8, 3)

(9, 2)

, or

(10, 1)

Minimizing the perimeter

P = x + y + 10

is equivalent to minimizing the sum

x + y

for integer values of

y > |x - 10|

Sum the possible integer values of

y

that yield the minimum perimeter.

3 + 2 + 1 = 6

We need to find the sum of all possible values of

y

that result in the minimum perimeter of

21

Key Concept

Triangle Inequality Theorem and Perimeter Optimization

Estimated Time:3m 0s

Question

Answer

Step-by-Step Solution

Key Concept

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