Question

an artist designs a rectangular quilt piece with different types of ribbon that go from the corner to the center of the quilt. the dimensions of the rectangle are ab = 10 inches and ac = 14 inches. find bx. image of rectangle with diagonals intersecting at x options: bx = 14 inches; bx = 5 inches; bx = 10 inches; bx = 7 inches. 10 multiple choice 9 points trsu is a rhombus. find su. image of rhombus trsu with side rt labeled 5x + 2 and side rs labeled 2x + 5 options: su = 5; su = 7; su = 1; su = 3. 11 true or false 9 points in a parallelogram, consecutive angles are congruent. options: true; false

Explanation:

First Question (Rectangle)

Step1: Recall Rectangle Diagonals Property

In a rectangle, diagonals are equal and bisect each other. So \( AC = BD \) and \( AX = CX = BX = DX=\frac{AC}{2}\) (or \(\frac{BD}{2}\)).

Step2: Calculate \( BX \)

Given \( AC = 14 \) inches. Since diagonals bisect each other, \( BX=\frac{AC}{2}=\frac{14}{2} = 7 \) inches.

Step1: Recall Rhombus Sides Property

In a rhombus, all sides are equal. So \( RT = RS \). Given \( RT = 5x + 2 \) and \( RS = 2x + 5 \).

Step2: Solve for \( x \)

Set \( 5x + 2=2x + 5 \). Subtract \( 2x \) from both sides: \( 3x+2 = 5 \). Subtract 2: \( 3x = 3 \). Divide by 3: \( x = 1 \).

Step3: Find \( SU \) (Note: Wait, in rhombus \( TRSU \), \( SU \) is a side? Wait, actually, sides of rhombus are equal, so \( SU = RT \) (or \( RS \)). Substitute \( x = 1 \) into \( RS = 2x + 5 \): \( 2(1)+5=7 \)? Wait, no, wait the options are 5,7,1,3. Wait, maybe I misread. Wait, \( TR = 5x + 2 \), \( RS = 2x + 5 \). So when \( x = 1 \), \( TR = 5(1)+2 = 7 \), \( RS = 2(1)+5 = 7 \). But the options for \( SU \): wait, maybe \( SU \) is equal to \( RS \)? Wait, no, maybe the problem has a typo, but according to calculation, when \( x = 1 \), side length is 7, but options have 7? Wait no, the options are SU = 5,7,1,3. Wait, maybe \( SU \) is \( RS \), so when \( x = 1 \), \( RS = 2(1)+5 = 7 \), but option has SU = 7? Wait the options are SU = 5, SU = 7, SU = 1, SU = 3. So if \( x = 1 \), \( RS = 7 \), so \( SU = 7 \)? But let's check again. Wait, rhombus sides: \( TR = RS = SU = UT \). So \( 5x + 2 = 2x + 5 \) gives \( x = 1 \), so \( RS = 2(1)+5 = 7 \), so \( SU = 7 \)? But the options include SU = 7? Wait the options are:

• SU = 5
• SU = 7
• SU = 1
• SU = 3

So the answer should be SU = 7? Wait but let's re - check. Wait, maybe I made a mistake. Wait, \( 5x + 2=2x + 5 \)
\( 5x-2x=5 - 2 \)
\( 3x = 3 \)
\( x = 1 \)
Then \( RS = 2(1)+5 = 7 \), so \( SU = RS = 7 \).

Step1: Recall Parallelogram Angles Property

In a parallelogram, consecutive angles are supplementary (sum to \( 180^{\circ} \)), not congruent (unless it's a rectangle). So the statement "In a parallelogram, consecutive angles are congruent" is false.

Answer:

BX = 7 inches

Second Question (Rhombus)