EXEHCRCE 1 posints $$ A(3.8) $$, B $$ (-6,4) $$ nt $$ C(a,-7) $$ 1. Calenier a pour que le linnghe Afri wit rockingte ewn 06 3. Soit. $$ P $$ le symetrigue de B par mopputt a M G. Geluler lea coordonnees de $$ D $$ 4. Quethe est In Hature du quandrlatóre ADCD? On justiflera la réponce sama effuct wer le moindre calcal 5. Calculer l'aire du quarinilatere $$ A B C D $$ ot win perimetere (on donnera ce resist sout bis forme $$ a \sqrt{b}, a $$ ef bentiers. b le plus petit. possible) 6. a) Développer, réduire et ordonner $$ (z-6)(4 z+19) $$.

Question

UpStudy AI · Accepted Answer

Let's break down the problem step by step.

### 1. Find $$ a $$ such that the line $$ AB $$ is parallel to the line $$ AC $$.

**Known Points:**
- $$ A(3, 8) $$
- $$ B(-6, 4) $$
- $$ C(a, -7) $$

**Step 1: Calculate the slope of line $$ AB $$.**
The slope $$ m_{AB} $$ is given by the formula:
$$
m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 8}{-6 - 3} = \frac{-4}{-9} = \frac{4}{9}
$$

**Step 2: Calculate the slope of line $$ AC $$.**
The slope $$ m_{AC} $$ is given by:
$$
m_{AC} = \frac{-7 - 8}{a - 3} = \frac{-15}{a - 3}
$$

**Step 3: Set the slopes equal for parallel lines.**
Since $$ AB $$ is parallel to $$ AC $$:
$$
\frac{4}{9} = \frac{-15}{a - 3}
$$

**Step 4: Cross-multiply and solve for $$ a $$.**
$$
4(a - 3) = -135
$$
$$
4a - 12 = -135
$$
$$
4a = -123
$$
$$
a = -\frac{123}{4} = -30.75
$$

### 2. Find the coordinates of point $$ D $$, the symmetric of $$ B $$ with respect to point $$ M $$.

**Step 1: Find the midpoint $$ M $$ of segment $$ AB $$.**
The coordinates of $$ M $$ are given by:
$$
M\left(\frac{x_A + x_B}{2}, \frac{y_A + y_B}{2}\right) = M\left(\frac{3 + (-6)}{2}, \frac{8 + 4}{2}\right) = M\left(-\frac{3}{2}, 6\right)
$$

**Step 2: Find the coordinates of point $$ D $$.**
The coordinates of $$ D $$ can be found using the midpoint formula:
$$
M = \left(\frac{x_B + x_D}{2}, \frac{y_B + y_D}{2}\right)
$$
Setting up the equations:
$$
-\frac{3}{2} = \frac{-6 + x_D}{2} \quad \text{and} \quad 6 = \frac{4 + y_D}{2}
$$

**Solving for $$ x_D $$ and $$ y_D $$:**
1. For $$ x_D $$:
$$
-\frac{3}{2} = \frac{-6 + x_D}{2} \implies -3 = -6 + x_D \implies x_D = 3
$$

2. For $$ y_D $$:
$$
6 = \frac{4 + y_D}{2} \implies 12 = 4 + y_D \implies y_D = 8
$$

Thus, $$ D(3, 8) $$.

### 3. Determine the nature of quadrilateral $$ ABCD $$.

**Step 1: Check if $$ A, B, C, D $$ are vertices of a parallelogram.**
To check if $$ ABCD $$ is a parallelogram, we can check if the midpoints of the diagonals $$ AC $$ and $$ BD $$ are the same.

**Midpoint of $$ AC $$:**
$$
M_{AC} = \left(\frac{3 + a}{2}, \frac{8 + (-7)}{2}\right) = \left(\frac{3 + a}{2}, \frac{1}{2}\right)
$$

**Midpoint of $$ BD $$:**
$$
M_{BD} = \left(\frac{-6 + 3}{2}, \frac{4 + 8}{2}\right) = \left(-\frac{3}{2}, 6\right)
$$

Since the midpoints are not equal, $$ ABCD $$ is not a parallelogram.

### 4. Calculate the area of quadrilateral $$ ABCD $$ and its perimeter.

**Step 1: Use the shoelace formula for area.**
The area $$ A $$ of quadrilateral $$ ABCD $$ can be calculated using the formula:
$$
A = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right|
$$
Substituting the coordinates:
- $$ A(3, 8) $$
- $$ B(-6, 4) $$
- $$ C(-30.75, -7) $$
- $$ D(3, 8) $$

Calculating the area:
$$
A = \frac{1}{2} \left| 3 \cdot 4 + (-6) \cdot (-7) + (-30.75) \cdot 8 + 3 \cdot 8 - (8 \cdot (-6) + 4 \cdot (-30.75) + (-7) \cdot 3 + 8 \cdot 3) \right|
$$

**Step 2: Calculate the perimeter.**
The perimeter $$ P $$ is the sum of the lengths of the sides:
$$
P = AB + BC + CD + DA
$$
Using the distance formula:
$$
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$

### 5. Develop, reduce, and order $$ (z-6)(4z+19) $$.

**Step 1: Expand the expression.**
$$
(z - 6)(4z + 19) = z \cdot 4z + z \cdot 19 - 6 \cdot 4z - 6 \cdot 19
$$
$$
= 4z^2 + 19z - 24z - 114
$$
$$
= 4z^2 - 5z - 114
$$

Now, let's calculate the area and perimeter of quadrilateral $$ ABCD $$ and provide the final results.

### Final Calculations
I will now calculate the area and perimeter using the shoelace formula and distance formula.
Calculate the value by following steps:
- step0: Calculate:
  $$3\times 4+\left(-6\right)\left(-7\right)+\left(-30\right)$$
- step1: Remove the parentheses:
  $$3\times 4-6\left(-7\right)-30$$
- step2: Multiply the numbers:
  $$12-6\left(-7\right)-30$$
- step3: Multiply the numbers:
  $$12+42-30$$
- step4: Calculate:
  $$24$$
Calculate or simplify the expression $$ (3*4 + (-6)*(-7) + (-30.75)*8 + 3*8) - (8*(-6) + 4*(-30.75) + (-7)*3 + 8*3) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(3\times 4+\left(-6\right)\left(-7\right)+\left(-30.75\right)\times 8+3\times 8\right)-\left(8\left(-6\right)+4\left(-30.75\right)+\left(-7\right)\times 3+8\times 3\right)$$
- step1: Remove the parentheses:
  $$\left(3\times 4-6\left(-7\right)-30.75\times 8+3\times 8\right)-\left(8\left(-6\right)+4\left(-30.75\right)+\left(-7\right)\times 3+8\times 3\right)$$
- step2: Multiply the numbers:
  $$\left(12-6\left(-7\right)-30.75\times 8+3\times 8\right)-\left(8\left(-6\right)+4\left(-30.75\right)+\left(-7\right)\times 3+8\times 3\right)$$
- step3: Multiply the numbers:
  $$\left(12+42-30.75\times 8+3\times 8\right)-\left(8\left(-6\right)+4\left(-30.75\right)+\left(-7\right)\times 3+8\times 3\right)$$
- step4: Multiply the numbers:
  $$\left(12+42-246+3\times 8\right)-\left(8\left(-6\right)+4\left(-30.75\right)+\left(-7\right)\times 3+8\times 3\right)$$
- step5: Multiply the numbers:
  $$\left(12+42-246+24\right)-\left(8\left(-6\right)+4\left(-30.75\right)+\left(-7\right)\times 3+8\times 3\right)$$
- step6: Calculate:
  $$\left(-168\right)-\left(8\left(-6\right)+4\left(-30.75\right)+\left(-7\right)\times 3+8\times 3\right)$$
- step7: Remove the parentheses:
  $$-168-\left(8\left(-6\right)+4\left(-30.75\right)-7\times 3+8\times 3\right)$$
- step8: Multiply the numbers:
  $$-168-\left(-48+4\left(-30.75\right)-7\times 3+8\times 3\right)$$
- step9: Multiply the numbers:
  $$-168-\left(-48-123-7\times 3+8\times 3\right)$$
- step10: Multiply the numbers:
  $$-168-\left(-48-123-21+8\times 3\right)$$
- step11: Multiply the numbers:
  $$-168-\left(-48-123-21+24\right)$$
- step12: Calculate:
  $$-168-\left(-168\right)$$
- step13: Remove the parentheses:
  $$-168+168$$
- step14: Add the numbers:
  $$0$$
Calculate or simplify the expression $$ \sqrt((3 - (-6))^2 + (8 - 4)^2) + \sqrt((-6 - (-30.75))^2 + (4 - (-7))^2) + \sqrt((-30.75 - 3)^2 + (-7 - 8)^2) + \sqrt((3 - 3)^2 + (8 - 8)^2) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\sqrt{\left(3-\left(-6\right)\right)^{2}+\left(8-4\right)^{2}}+\sqrt{\left(-6-\left(-30.75\right)\right)^{2}+\left(4-\left(-7\right)\right)^{2}}+\sqrt{\left(-30.75-3\right)^{2}+\left(-7-8\right)^{2}}+\sqrt{\left(3-3\right)^{2}+\left(8-8\right)^{2}}$$
- step1: Subtract the terms:
  $$\sqrt{\left(3-\left(-6\right)\right)^{2}+\left(8-4\right)^{2}}+\sqrt{\left(-6-\left(-30.75\right)\right)^{2}+\left(4-\left(-7\right)\right)^{2}}+\sqrt{\left(-30.75-3\right)^{2}+\left(-7-8\right)^{2}}+\sqrt{\left(3-3\right)^{2}+0^{2}}$$
- step2: Subtract the terms:
  $$\sqrt{\left(3-\left(-6\right)\right)^{2}+\left(8-4\right)^{2}}+\sqrt{\left(-6-\left(-30.75\right)\right)^{2}+\left(4-\left(-7\right)\right)^{2}}+\sqrt{\left(-30.75-3\right)^{2}+\left(-7-8\right)^{2}}+\sqrt{0^{2}+0^{2}}$$
- step3: Remove the parentheses:
  $$\sqrt{\left(3-\left(-6\right)\right)^{2}+\left(8-4\right)^{2}}+\sqrt{\left(-6-\left(-30.75\right)\right)^{2}+\left(4+7\right)^{2}}+\sqrt{\left(-30.75-3\right)^{2}+\left(-7-8\right)^{2}}+\sqrt{0^{2}+0^{2}}$$
- step4: Remove the parentheses:
  $$\sqrt{\left(3-\left(-6\right)\right)^{2}+\left(8-4\right)^{2}}+\sqrt{\left(-6+30.75\right)^{2}+\left(4+7\right)^{2}}+\sqrt{\left(-30.75-3\right)^{2}+\left(-7-8\right)^{2}}+\sqrt{0^{2}+0^{2}}$$
- step5: Remove the parentheses:
  $$\sqrt{\left(3+6\right)^{2}+\left(8-4\right)^{2}}+\sqrt{\left(-6+30.75\right)^{2}+\left(4+7\right)^{2}}+\sqrt{\left(-30.75-3\right)^{2}+\left(-7-8\right)^{2}}+\sqrt{0^{2}+0^{2}}$$
- step6: Add the numbers:
  $$\sqrt{9^{2}+\left(8-4\right)^{2}}+\sqrt{\left(-6+30.75\right)^{2}+\left(4+7\right)^{2}}+\sqrt{\left(-30.75-3\right)^{2}+\left(-7-8\right)^{2}}+\sqrt{0^{2}+0^{2}}$$
- step7: Subtract the numbers:
  $$\sqrt{9^{2}+4^{2}}+\sqrt{\left(-6+30.75\right)^{2}+\left(4+7\right)^{2}}+\sqrt{\left(-30.75-3\right)^{2}+\left(-7-8\right)^{2}}+\sqrt{0^{2}+0^{2}}$$
- step8: Add the numbers:
  $$\sqrt{9^{2}+4^{2}}+\sqrt{24.75^{2}+\left(4+7\right)^{2}}+\sqrt{\left(-30.75-3\right)^{2}+\left(-7-8\right)^{2}}+\sqrt{0^{2}+0^{2}}$$
- step9: Add the numbers:
  $$\sqrt{9^{2}+4^{2}}+\sqrt{24.75^{2}+11^{2}}+\sqrt{\left(-30.75-3\right)^{2}+\left(-7-8\right)^{2}}+\sqrt{0^{2}+0^{2}}$$
- step10: Subtract the numbers:
  $$\sqrt{9^{2}+4^{2}}+\sqrt{24.75^{2}+11^{2}}+\sqrt{\left(-33.75\right)^{2}+\left(-7-8\right)^{2}}+\sqrt{0^{2}+0^{2}}$$
- step11: Subtract the numbers:
  $$\sqrt{9^{2}+4^{2}}+\sqrt{24.75^{2}+11^{2}}+\sqrt{\left(-33.75\right)^{2}+\left(-15\right)^{2}}+\sqrt{0^{2}+0^{2}}$$
- step12: Add the numbers:
  $$\sqrt{97}+\sqrt{24.75^{2}+11^{2}}+\sqrt{\left(-33.75\right)^{2}+\left(-15\right)^{2}}+\sqrt{0^{2}+0^{2}}$$
- step13: Convert the expressions:
  $$\sqrt{97}+\sqrt{\left(\frac{99}{4}\right)^{2}+11^{2}}+\sqrt{\left(-33.75\right)^{2}+\left(-15\right)^{2}}+\sqrt{0^{2}+0^{2}}$$
- step14: Add the numbers:
  $$\sqrt{97}+\sqrt{\frac{11737}{16}}+\sqrt{\left(-33.75\right)^{2}+\left(-15\right)^{2}}+\sqrt{0^{2}+0^{2}}$$
- step15: Simplify the root:
  $$\sqrt{97}+\frac{11\sqrt{97}}{4}+\sqrt{\left(-33.75\right)^{2}+\left(-15\right)^{2}}+\sqrt{0^{2}+0^{2}}$$
- step16: Convert the expressions:
  $$\sqrt{97}+\frac{11\sqrt{97}}{4}+\sqrt{\left(-\frac{135}{4}\right)^{2}+\left(-15\right)^{2}}+\sqrt{0^{2}+0^{2}}$$
- step17: Add the numbers:
  $$\sqrt{97}+\frac{11\sqrt{97}}{4}+\sqrt{\frac{21825}{16}}+\sqrt{0^{2}+0^{2}}$$
- step18: Simplify the root:
  $$\sqrt{97}+\frac{11\sqrt{97}}{4}+\frac{15\sqrt{97}}{4}+\sqrt{0^{2}+0^{2}}$$
- step19: Calculate:
  $$\sqrt{97}+\frac{11\sqrt{97}}{4}+\frac{15\sqrt{97}}{4}+\sqrt{0+0^{2}}$$
- step20: Calculate:
  $$\sqrt{97}+\frac{11\sqrt{97}}{4}+\frac{15\sqrt{97}}{4}+\sqrt{0+0}$$
- step21: Remove 0:
  $$\sqrt{97}+\frac{11\sqrt{97}}{4}+\frac{15\sqrt{97}}{4}+\sqrt{0}$$
- step22: Simplify the root:
  $$\sqrt{97}+\frac{11\sqrt{97}}{4}+\frac{15\sqrt{97}}{4}+0$$
- step23: Remove 0:
  $$\sqrt{97}+\frac{11\sqrt{97}}{4}+\frac{15\sqrt{97}}{4}$$
- step24: Reduce fractions to a common denominator:
  $$\frac{\sqrt{97}\times 4}{4}+\frac{11\sqrt{97}}{4}+\frac{15\sqrt{97}}{4}$$
- step25: Transform the expression:
  $$\frac{\sqrt{97}\times 4+11\sqrt{97}+15\sqrt{97}}{4}$$
- step26: Reorder the terms:
  $$\frac{4\sqrt{97}+11\sqrt{97}+15\sqrt{97}}{4}$$
- step27: Add the numbers:
  $$\frac{30\sqrt{97}}{4}$$
- step28: Reduce the fraction:
  $$\frac{15\sqrt{97}}{2}$$
### Summary of Calculations

1. **Finding $$ a $$**:
   - We found $$ a = -30.75 $$ such that line $$ AB $$ is parallel to line $$ AC $$.

2. **Coordinates of Point $$ D $$**:
   - The coordinates of point $$ D $$ are $$ D(3, 8) $$.

3. **Nature of Quadrilateral $$ ABCD $$**:
   - The quadrilateral $$ ABCD $$ is not a parallelogram since the midpoints of the diagonals $$ AC $$ and $$ BD $$ are not equal.

4. **Area of Quadrilateral $$ ABCD $$**:
   - The area calculated using the shoelace formula is $$ 0 $$. This indicates that points $$ A, B, C, D $$ are collinear.

5. **Perimeter of Quadrilateral $$ ABCD $$**:
   - The perimeter is calculated as $$ \frac{15\sqrt{97}}{2} $$.

6. **Expansion of $$ (z-6)(4z+19) $$**:
   - The expanded form is $$ 4z^2 - 5z - 114 $$.

### Final Results
- $$ a = -30.75 $$
- Coordinates of $$ D = (3, 8) $$
- Area of quadrilateral $$ ABCD = 0 $$
- Perimeter of quadrilateral $$ ABCD = \frac{15\sqrt{97}}{2} $$
- Expanded form of $$ (z-6)(4z+19) = 4z^2 - 5z - 114 $$
- $$ a = -30.75 $$ - Coordinates of $$ D = (3, 8) $$ - Area of quadrilateral $$ ABCD = 0 $$ - Perimeter of quadrilateral $$ ABCD = \frac{15\sqrt{97}}{2} $$ - Expanded form of $$ (z-6)(4z+19) = 4z^2 - 5z - 114 $$

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