Question

in exercises 9 - 14, use the diagram.

1. find the perimeter of △abd
2. find the perimeter of △bcd
3. find the perimeter of quadrilateral abcd
4. find the area of △abd
5. find the area of △bcd
6. find the area of quadrilateral abcd

Explanation:

Step1: Find side - lengths using distance formula

For two points $(x_1,y_1)$ and $(x_2,y_2)$, distance $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$. When $x$-coordinates are the same, distance is $\vert y_2 - y_1\vert$; when $y$-coordinates are the same, distance is $\vert x_2 - x_1\vert$.

Step2: Calculate perimeters

Sum up the side - lengths of the triangles or quadrilateral.

Step3: Calculate areas of triangles

Use the formula $A=\frac{1}{2}\times base\times height$. Identify the base (a horizontal or vertical side if possible) and the corresponding height.

Step4: Calculate area of quadrilateral

Sum up the areas of the two non - overlapping triangles that make up the quadrilateral.

Answer:

1. Perimeter of $\triangle ABD$:

Let $A(2,4)$, $B(2, - 1)$, $D(-4,-1)$.
First, find the length of $AB$:
Since the $x$-coordinates of $A$ and $B$ are the same ($x = 2$), $AB=\vert4-(-1)\vert = 5$.
Next, find the length of $BD$:
Since the $y$-coordinates of $B$ and $D$ are the same ($y=-1$), $BD=\vert2 - (-4)\vert=6$.
Then, find the length of $AD$ using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$
$x_1 = 2,y_1 = 4,x_2=-4,y_2=-1$
$AD=\sqrt{( - 4 - 2)^2+( - 1 - 4)^2}=\sqrt{(-6)^2+( - 5)^2}=\sqrt{36 + 25}=\sqrt{61}\approx 7.81$
Perimeter of $\triangle ABD=AB + BD+AD=5 + 6+\sqrt{61}\approx18.81$

1. Perimeter of $\triangle BCD$:

Let $B(2,-1)$, $C(-1,-5)$, $D(-4,-1)$
Length of $BC$: $d_{BC}=\sqrt{( - 1 - 2)^2+( - 5+1)^2}=\sqrt{(-3)^2+( - 4)^2}=\sqrt{9 + 16}=5$
Length of $CD$: Since the $y$-coordinates of $C$ and $D$ are the same ($y = - 1$), $CD=\vert-4-( - 1)\vert=3$
Length of $BD$: Since the $y$-coordinates of $B$ and $D$ are the same ($y=-1$), $BD=\vert2-( - 4)\vert = 6$
Perimeter of $\triangle BCD=BC + CD+BD=5 + 3+6 = 14$

1. Perimeter of quadrilateral $ABCD$:

We already know some side - lengths from previous calculations.
We need to find the length of $AC$:
$A(2,4)$, $C(-1,-5)$
$AC=\sqrt{( - 1 - 2)^2+( - 5 - 4)^2}=\sqrt{(-3)^2+( - 9)^2}=\sqrt{9 + 81}=\sqrt{90}=3\sqrt{10}\approx9.49$
Perimeter of $ABCD=AB + BC+CD + DA$
$AB = 5$, $BC = 5$, $CD = 3$, $DA=\sqrt{61}\approx7.81$
Perimeter$=5 + 5+3+\sqrt{61}\approx20.81$

1. Area of $\triangle ABD$:

Base $BD = 6$ and height (vertical distance from $A$ to $BD$) $h = 5$
Area of $\triangle ABD=\frac{1}{2}\times base\times height=\frac{1}{2}\times6\times5 = 15$

1. Area of $\triangle BCD$:

Base $BD = 6$ and height (vertical distance from $C$ to $BD$) $h = 4$
Area of $\triangle BCD=\frac{1}{2}\times base\times height=\frac{1}{2}\times6\times4 = 12$

1. Area of quadrilateral $ABCD$:

Area of $ABCD=$ Area of $\triangle ABD+$ Area of $\triangle BCD=15 + 12=27$