A triangle has two sides of length 8 and 15. If the length of the third side, s, is a prim… — ACT

A triangle has two sides of length $8$ and $15$ . If the length of the third side, $s$ , is a prime number, how many possible values are there for $s$ ?

A
$3$
$4$ Answer
C
$5$
D
$6$
E
$7$

Answer

There are 4 possible values for the third side length s.

The correct answer is

4

. According to the Triangle Inequality Theorem, the length of the third side

s

of a triangle with sides of

8

and

15

must satisfy

15 - 8 < s < 15 + 8

. This simplifies to the open interval

7 < s < 23

. The prime numbers strictly between

7

and

23

are

11

13

17

, and

19

. Counting these gives exactly

4

possible prime values for

s

Step-by-Step Solution

Apply the Triangle Inequality Theorem to determine the bounds for the third side

s

15 - 8 < s < 15 + 8

, which simplifies to

7 < s < 23

The Triangle Inequality Theorem states that the length of any side of a triangle must be strictly greater than the difference of the other two sides and strictly less than their sum.

Identify all prime numbers that lie strictly within the range

(7, 23)

The prime numbers in this range are

11

13

17

, and

19

A prime number is an integer greater than 1 that has no positive divisors other than 1 and itself.

Count the number of identified prime numbers.

There are

4

prime numbers (

11

13

17

19

This counts the total number of possible valid lengths for

s

Key Concept

Triangle Inequality Theorem and basic number properties

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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