Question

7 multiple choice 9 points
in the diagram below of parallelogram stuv, sv = x + 3, vu = 2x - 1, and tu = 4x - 3.
what is the length of \\(\overline{sv}\\)?
\\(\bigcirc\\) 4
\\(\bigcirc\\) 7
\\(\bigcirc\\) 5
\\(\bigcirc\\) 2
8 multiple choice 9 points
a quadrilateral whose diagonals bisect each other and are perpendicular is a \\_\\_\\_\\_\\_\\_\\_.
\\(\bigcirc\\) parallelogram
\\(\bigcirc\\) rectangle
\\(\bigcirc\\) rhombus
\\(\bigcirc\\) trapezoid

Explanation:

Question 7

Step1: Recall parallelogram property

In a parallelogram, opposite sides are equal. So, \( SV = TU \) (wait, no, wait: Wait, in parallelogram \( STUV \), \( SV \) and \( TU \)? Wait, no, let's check the sides. Wait, \( SV \) and \( TU \)? Wait, no, actually, in a parallelogram, opposite sides are equal. So \( SV = TU \)? Wait, no, wait the labels: \( S, T, U, V \). So the sides: \( SV \) and \( TU \)? Wait, no, maybe \( SV = TU \)? Wait, no, let's see: \( SV \) is from \( S \) to \( V \), \( TU \) is from \( T \) to \( U \). Wait, maybe I made a mistake. Wait, in a parallelogram, opposite sides are equal. So \( SV = TU \)? Wait, no, wait \( SV \) and \( TU \): Wait, \( SV \) and \( TU \) – no, actually, \( SV \) and \( TU \) are opposite sides? Wait, no, let's list the sides: \( STUV \) is a parallelogram, so \( SV \parallel TU \) and \( ST \parallel VU \). So opposite sides: \( SV = TU \)? Wait, no, \( SV \) and \( TU \) – wait, \( SV \) is length \( x + 3 \), \( TU \) is \( 4x - 3 \). Wait, but also \( VU = 2x - 1 \), and \( ST = VU \) (since opposite sides of parallelogram are equal). Wait, maybe I confused the sides. Wait, the problem says \( SV = x + 3 \), \( VU = 2x - 1 \), \( TU = 4x - 3 \). In a parallelogram, opposite sides are equal. So \( SV = TU \)? Wait, no, \( SV \) and \( TU \) – wait, \( SV \) is adjacent to \( VU \), so \( SV \) and \( TU \) are opposite? Wait, maybe \( SV = TU \)? Wait, let's check: If \( SV = TU \), then \( x + 3 = 4x - 3 \). Solving: \( 3 + 3 = 4x - x \) → \( 6 = 3x \) → \( x = 2 \). Then \( SV = x + 3 = 2 + 3 = 5 \). Wait, but let's check \( VU = 2x - 1 = 2*2 -1 = 3 \), and \( TU = 4x -3 = 5 \). Then \( SV = 5 \), \( TU = 5 \), \( VU = 3 \), \( ST = 3 \). That works. Wait, but let's confirm the property. In a parallelogram, opposite sides are equal. So \( SV = TU \) (since \( SV \) and \( TU \) are opposite sides). So set \( SV = TU \): \( x + 3 = 4x - 3 \). Solve for \( x \):

Step2: Solve for \( x \)

\( x + 3 = 4x - 3 \)

Subtract \( x \) from both sides: \( 3 = 3x - 3 \)

Add 3 to both sides: \( 6 = 3x \)

Divide by 3: \( x = 2 \)

Step3: Find length of \( SV \)

\( SV = x + 3 = 2 + 3 = 5 \)

A quadrilateral with diagonals that bisect each other is a parallelogram (by the theorem: If diagonals of a quadrilateral bisect each other, it's a parallelogram). Now, among parallelograms, those with perpendicular diagonals are rhombuses (theorem: A parallelogram with perpendicular diagonals is a rhombus). A rectangle has diagonals equal but not necessarily perpendicular (unless it's a square). A trapezoid has only one pair of parallel sides, and its diagonals don't necessarily bisect each other. So the answer is rhombus.

Answer:

5

Question 8