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Question

Question
Rhombus QRST has diagonal lines QS¯¯¯¯¯¯¯ and RT¯¯¯¯¯¯¯ that intersect at U . If QS¯¯¯¯¯¯¯=30 and RT¯¯¯¯¯¯¯=16 , what is the length of one of the sides of the rhombus?(1 point)
Responses which one is correct and why

46
46

14
14

34
34

17
17
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Answer 1

To find the length of one side of the rhombus QRST, we can use the properties of rhombuses and the Pythagorean theorem.

In a rhombus, the diagonals bisect each other at right angles. This means that point U is the midpoint of both diagonals QS and RT. Given the lengths of the diagonals:

• QS = 30, so each half (QU and US) = 30/2 = 15.
• RT = 16, so each half (RU and UT) = 16/2 = 8.

Now we can use these segments to form a right triangle, where one side is 15 (QU) and the other side is 8 (RU). Let’s denote the length of one side of the rhombus as \( s \).

Using the Pythagorean theorem:

\[ s^2 = QU^2 + RU^2 \]

Substituting in the lengths:

\[ s^2 = 15^2 + 8^2 \]

Calculating the squares:

\[ s^2 = 225 + 64 \] \[ s^2 = 289 \]

Now taking the square root:

\[ s = \sqrt{289} = 17 \]

Thus, the length of one side of the rhombus is 17.

The correct response is 17.