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

46
46

14
14

17
17

34
34
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Answer 1

To find the length of one side of the rhombus QRST, we can utilize the properties of the diagonals in a rhombus. The diagonals of a rhombus bisect each other at right angles.

Given:
- Length of diagonal \( QS = 30 \)
- Length of diagonal \( RT = 16 \)

First, we find the lengths of the segments created by the intersection of the diagonals at point U:
- Since diagonal \( QS \) is 30, each half will be:
\[
QU = \frac{30}{2} = 15
\]
- Since diagonal \( RT \) is 16, each half will be:
\[
RU = \frac{16}{2} = 8
\]

Now, triangle QUR is a right triangle, where:
- \( QU = 15 \)
- \( RU = 8 \)

To find the length of one side of the rhombus (let's call it \( QR \)), we can use the Pythagorean theorem:
\[
QR^2 = QU^2 + RU^2
\]
Substituting the values we found:
\[
QR^2 = 15^2 + 8^2
\]
Calculating this:
\[
QR^2 = 225 + 64 = 289
\]
Taking the square root of both sides to find QR:
\[
QR = \sqrt{289} = 17
\]

Therefore, the length of one of the sides of the rhombus is **17**.