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Question

Parallelogram upper R upper S upper T upper U is shown with base upper T upper U larger than height upper S upper T. Diagonals are drawn from upper R to upper T and upper S to upper U to meet at upper V.

In parallelogram RSTU
, RV
measures y+18
units and TV
measures 3y−10
units. How many units long is diagonal RT
?

(1 point)
Responses

64
64

16
16

14
14

32

Answer 1

In a parallelogram, the diagonals bisect each other. Therefore, the lengths of segments from the vertices to the intersection point are half of the lengths of the diagonals.

Given:

• \( RV = y + 18 \)
• \( TV = 3y - 10 \)

Since \( RV = TV \), we can set up the equation:

\[ y + 18 = 3y - 10 \]

Now, solve for \( y \):

1. Rearranging the equation gives: \[ 18 + 10 = 3y - y \] \[ 28 = 2y \]

2. Divide both sides by 2: \[ y = 14 \]

Now substitute \( y \) back into the expressions for \( RV \) and \( TV \):

\[ RV = 14 + 18 = 32 \] \[ TV = 3(14) - 10 = 42 - 10 = 32 \]

Since both segment lengths equal 32, we find \( RT \) (the whole diagonal) by doubling the length of either segment:

\[ RT = RV + TV = RV + RV = 32 + 32 = 64 \]

Thus, the length of diagonal \( RT \) is:

\[ \boxed{64} \]