12. multiple choice in parallelogram abcd, $ab = 2x$, $ac = 3x - 2$, and $ad = x + 2$. if the length of segment $bd$ is 10, what value of $x$ will ensure that abcd is a rectangle? (lesson 7-4)a. 2b. 4c. 5d. 813. multi-select if four bars of equal length are joined at their endpoints, which shape(s) can be created? (lesson 7-5)a. kiteb. parallelogramc. rectangled. rhombuse. squaref. trapezoid14. multiple choice find the measure of $angle b$. (lesson 7-5)a. $40^circ$b. $50^circ$c. $80^circ$d. $100^circ$15. multi-select if abcd is a rhombus that is not a square, select all of the true statements. (lesson 7-5)a. $overline{ab} cong overline{cd}$b. $overline{ac} perp overline{bd}$c. $angle a cong angle c$d. $overline{ac} cong overline{bd}$e. $overline{bc} cong overline{da}$16. open response given quadrilateral jklm with $j(-12, 0)$, $k(10, 0)$, $l(6, 0)$, and $m(0, -5)$, how can it be determined whether the quadrilateral is a kite?17. multiple choice trapezoid abcd has vertices $a(0, 0)$, $b(2, 5)$, $c(3, 5)$, and $d(8, 0)$. what is the length of its midsegment? (lesson 7-6)a. 4b. 4.5c. 5d. 5.5

Question

Accepted Answer

### Question 12
# Explanation:
## Step1: Recall rectangle diagonal property
In a rectangle, diagonals are equal, so $AC = BD$. For a parallelogram, $BD = 2 	imes$ half-diagonal, but here we use the side-length relationship for a rectangle: by the Pythagorean theorem, $AB^2 + AD^2 = AC^2$.
## Step2: Substitute expressions
Substitute $AB=2x$, $AD=x+2$, $AC=3x-2$:
$$(2x)^2 + (x+2)^2 = (3x-2)^2$$
## Step3: Expand all terms
$$4x^2 + x^2 +4x +4 = 9x^2 -12x +4$$
## Step4: Simplify equation
$$5x^2 +4x +4 -9x^2 +12x -4 = 0$$
$$-4x^2 +16x = 0$$
$$-4x(x-4)=0$$
## Step5: Solve for valid x
$x=0$ is invalid (side length can't be 0), so $x=4$.
# Answer:
B. 4

---

### Question 13
# Brief Explanations:
- A kite requires two distinct pairs of adjacent equal sides, so equal-length bars cannot form it.
- A parallelogram is formed when equal-length bars are joined as opposite sides.
- A rectangle needs right angles, which is not guaranteed by only equal bar lengths.
- A rhombus is a parallelogram with all sides equal, which fits equal-length bars.
- A square is a rhombus with right angles, which is a special case possible if joined at right angles.
- A trapezoid requires at least one pair of parallel sides, which can be formed with equal bars.
# Answer:
B. parallelogram, D. rhombus, E. square, F. trapezoid

---

### Question 14
# Explanation:
## Step1: Identify quadrilateral type
The figure is a rhombus (all sides marked equal). In a rhombus, consecutive angles are supplementary.
## Step2: Calculate $\angle B$
$\angle B + 100^\circ = 180^\circ$
$$\angle B = 180^\circ - 100^\circ = 80^\circ$$
# Answer:
C. $80^\circ$

---

### Question 15
# Brief Explanations:
- A. In all parallelograms (including rhombuses), opposite sides are congruent, so $\overline{AB} \cong \overline{CD}$ is true.
- B. All rhombuses have perpendicular diagonals, so $\overline{AC} \perp \overline{BD}$ is true.
- C. In all parallelograms, opposite angles are congruent, so $\angle A \cong \angle C$ is true.
- D. Rhombus diagonals are only congruent if it is a square, so $\overline{AC} \cong \overline{BD}$ is false.
- E. In all parallelograms, opposite sides are congruent, so $\overline{BC} \cong \overline{DA}$ is true.
# Answer:
A. $\overline{AB} \cong \overline{CD}$, B. $\overline{AC} \perp \overline{BD}$, C. $\angle A \cong \angle C$, E. $\overline{BC} \cong \overline{DA}$

---

### Question 16
# Brief Explanations:
A quadrilateral is a kite if it has exactly one pair of congruent opposite angles, or two distinct pairs of adjacent congruent sides. Using coordinates:
1. Calculate the length of all four sides using the distance formula $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.
2. Verify if there are two distinct pairs of adjacent sides with equal lengths, and no opposite sides are equal (to rule out a rhombus).
3. Alternatively, check if one diagonal is the perpendicular bisector of the other, but not vice versa.
# Answer:
1. Calculate the lengths of $\overline{JK}$, $\overline{KL}$, $\overline{LM}$, $\overline{MJ}$ using the distance formula $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.
2. Confirm that there are two pairs of adjacent sides with equal lengths (e.g., $JK=KL$ and $LM=MJ$, or $JK=MJ$ and $KL=LM$) and that opposite sides are not equal. If this is true, $JKLM$ is a kite.

---

### Question 17
# Explanation:
## Step1: Recall trapezoid midsegment formula
The length of a trapezoid midsegment is $\frac{1}{2} 	imes$ (sum of the lengths of the two parallel bases).
## Step2: Identify parallel bases
Points $A(0,0)$, $D(8,0)$ are on the x-axis; points $B(2,5)$, $C(3,5)$ are on the line $y=5$. So the bases are $\overline{AD}$ and $\overline{BC}$.
## Step3: Calculate base lengths
Length of $\overline{AD}$: $8-0=8$
Length of $\overline{BC}$: $3-2=1$
## Step4: Compute midsegment length
$$	ext{Midsegment length} = \frac{1}{2} 	imes (8+1) = 4.5$$
# Answer:
B. 4.5

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Question 12

Step1: Recall rectangle diagonal property

Step2: Substitute expressions

Step3: Expand all terms

Step4: Simplify equation

Step5: Solve for valid x

Step1: Identify quadrilateral type

Step2: Calculate $\angle B$

Answer:

Question 13