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 17 17 46 46 34 34 14

Answer 1

In a rhombus, the diagonals bisect each other at right angles. This means that point U, where the diagonals intersect, divides each diagonal into two equal parts.

Given the lengths of the diagonals QS and RT:

• $QS=30$ implies $QU=US=\frac{30}{2}=15$
• $RT=16$ implies $RU=UT=\frac{16}{2}=8$

Now, we can use the Pythagorean theorem to find the length of one side of the rhombus. Let the length of one side of the rhombus be $s$. In triangle QUR, we have:

$$s^2=Q{U}^2+R{U}^2$$

Substituting the values we found:

$$s^2={15}^2+8^2$$ $$s^2=225+64$$ $$s^2=289$$ $$s=\sqrt{289}=17$$

Thus, the length of one of the sides of the rhombus is $\boxed{{\displaystyle 17}}$.