Question

in the rectangle below, rv = 3x + 7, sv = 5x - 3, and m∠vst = 51°. find rt and m∠vtu.

Explanation:

Step1: Find x using rectangle diagonals property

In a rectangle, diagonals are equal and bisect each other. So \( RV = SV \).
Set \( 3x + 7 = 5x - 3 \).
Subtract \( 3x \) from both sides: \( 7 = 2x - 3 \).
Add 3 to both sides: \( 10 = 2x \).
Divide by 2: \( x = 5 \).

Step2: Calculate RV and SV

Substitute \( x = 5 \) into \( RV = 3x + 7 \): \( RV = 3(5) + 7 = 15 + 7 = 22 \).
Since \( RV = SV \), \( SV = 22 \).

Step3: Find RT (diagonal)

In a rectangle, diagonal \( RT = RV + VT \), and \( RV = VT \) (diagonals bisect each other). So \( RT = 2 \times RV = 2 \times 22 = 44 \).

Step4: Find \( m\angle VTU \)

In rectangle \( RSTU \), \( \angle VST = 51^\circ \), and \( \angle STU = 90^\circ \) (rectangle angle).
In \( \triangle VST \), \( \triangle VST \) is isosceles ( \( SV = TV \) as diagonals bisect and are equal), but we can use complementary angles.
\( \angle VTU = 90^\circ - 51^\circ = 39^\circ \).

Answer:

\( RT = 44 \)
\( m\angle VTU = 39^\circ \)