The side lengths of triangle ABC, in units, are given by AB = x, BC = x + 2, and CA = x + … — GRE General Test

The side lengths of triangle $ABC$ , in units, are given by $AB = x$ , $BC = x + 2$ , and $CA = x + 4$ , where $x$ is a positive integer. If the perimeter of triangle $ABC$ is strictly less than $36$ , how many possible integer values of $x$ exist such that triangle $ABC$ is an acute triangle?

A
2
3Answer
C
4
D
5
E
7

Answer

There are 3 possible integer values of

x

To form an acute triangle with side lengths

x

x+2

, and

x+4

, three conditions must be met: the perimeter bound (

3x+6 < 36 \implies x < 10

), the non-degeneracy condition (

x + x + 2 > x + 4 \implies x > 2

), and the acute angle condition (

(x+4)^2 < x^2 + (x+2)^2 \implies x^2 - 4x - 12 > 0 \implies x > 6

). Combining these bounds restricts

x

to integer values in the range

6 < x < 10

, giving exactly three valid integer values:

7, 8,

and

9

Step-by-Step Solution

Apply the perimeter constraint to find an upper bound for

x

The perimeter is

P = x + (x + 2) + (x + 4) = 3x + 6

. Given

3x + 6 < 36

, subtracting

6

yields

3x < 30

, so

x < 10

The total length around the triangle must be strictly less than

36

Apply the triangle inequality theorem to find a lower bound for non-degenerate triangles.

AB + BC > CA \implies x + (x + 2) > x + 4 \implies 2x + 2 > x + 4 \implies x > 2

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

Formulate and solve the acute triangle condition.

For an acute triangle with longest side

CA = x + 4

, we require

CA^2 < AB^2 + BC^2

(x + 4)^2 < x^2 + (x + 2)^2

x^2 + 8x + 16 < x^2 + x^2 + 4x + 4

x^2 + 8x + 16 < 2x^2 + 4x + 4

0 < x^2 - 4x - 12

(x - 6)(x + 2) > 0

Since

x

is a positive integer,

x + 2 > 0

, which implies

x - 6 > 0

, so

x > 6

In any acute triangle, the square of the longest side is strictly less than the sum of the squares of the remaining two sides.

Combine the bounds to determine the number of valid integer values for

x

Combining

x > 6

and

x < 10

gives integer values

x \in \{7, 8, 9\}

. There are

3

such values.

Only integer values in the open interval

(6, 10)

satisfy all conditions simultaneously.

Key Concept

Acute Triangle Criteria and Triangle Inequality

Question

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