A triangle has two sides of length 9 and 14. The length of the third side, s, is a multipl… — ACT

A triangle has two sides of length $9$ and $14$ . The length of the third side, $s$ , is a multiple of $4$ . What is the sum of all possible integer values for $s$ ?

A
48
B
60
56Answer
D
76
E
80

Answer

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

s

, of a triangle with sides of length 9 and 14 must satisfy the inequality

14 - 9 < s < 14 + 9

, which simplifies to

5 < s < 23

. The integer values within this range that are multiples of 4 are 8, 12, 16, and 20. Adding these values together yields a sum of 56.

Step-by-Step Solution

Apply the Triangle Inequality Theorem to determine the range of possible lengths for the third side.

The third side length,

s

, must satisfy the inequality:

14 - 9 < s < 14 + 9

, which simplifies to

5 < s < 23

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

Identify all integer values in the range

(5, 23)

that are multiples of 4.

The multiples of 4 that are strictly greater than 5 and strictly less than 23 are: 8, 12, 16, and 20.

We must find the integers within the bounds that can be divided by 4 with a remainder of 0.

Calculate the sum of the identified multiples of 4.

8 + 12 + 16 + 20 = 56

The question asks for the sum of all possible integer values for the third side length

s

Key Concept

Triangle Inequality Theorem

Estimated Time:1m 0s

Question

Answer

Step-by-Step Solution

Key Concept

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