Question

#1.) determine the perimeter of a square lphs where side lp has endpoints (-10,9) and (-6,6). hint: a square has 4 equal sides.
**#2.) find ik.
i ———— 12 ———— l
j k
←———— 49 ————→
←———— 31 ————→
**#3.) keepin’ it fresh! solve: 2x² - 2x = 40
**#4.) the endpoints of ad are a and d. point b is inbetween a and d but closer to a. point c is inbetween b and d. if ad = x², ab = 2x + 10, bc = x + 6 and cd = 3x determine the length of bd.
a.) sketch a line segment using the information above.
b.) solve for the segment length.

Explanation:

Step1: Calculate side - length of square

Use the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ for two points $(x_1,y_1)=(-10,9)$ and $(x_2,y_2)=(-6,6)$.
$d=\sqrt{(-6+10)^2+(6 - 9)^2}=\sqrt{4^2+( - 3)^2}=\sqrt{16 + 9}=\sqrt{25}=5$

Step2: Calculate perimeter of square

Since a square has 4 equal sides, and the side - length $s = 5$, the perimeter $P=4s$.
$P = 4\times5=20$

Step1: Find length of $\overline{IK}$

We know that $IL=49$ and $LK = 31$, and $IL=IK+LK$. So, $IK=IL - LK$.
$IK=49-31 = 18$

Step1: Rewrite the quadratic equation

Given $2x^{2}-2x = 40$, divide the entire equation by 2 to simplify: $x^{2}-x-20=0$.

Step2: Factor the quadratic equation

We factor $x^{2}-x - 20$ as $(x - 5)(x+4)=0$.

Step3: Solve for $x$

Set each factor equal to zero: $x - 5=0$ gives $x = 5$; $x+4=0$ gives $x=-4$.

Step1: Sketch the line segment

Draw a line and label the endpoints as $A$ and $D$. Mark a point $B$ between $A$ and $D$ closer to $A$, and a point $C$ between $B$ and $D$. Label $AB = 2x + 10$, $BC=x + 6$, $CD = 3x$, and $AD=x^{2}$.

Step2: Set up an equation

Since $AD=AB + BC+CD$, we have $x^{2}=(2x + 10)+(x + 6)+3x$.
$x^{2}=2x+10+x + 6+3x$
$x^{2}=6x + 16$
$x^{2}-6x - 16=0$

Step3: Factor the quadratic equation

Factor $x^{2}-6x - 16$ as $(x - 8)(x+2)=0$.

Step4: Solve for $x$

$x - 8=0$ gives $x = 8$; $x+2=0$ gives $x=-2$. But since lengths cannot be negative in this context, we take $x = 8$.

Step5: Find the length of $\overline{BD}$

$BD=BC + CD=(x + 6)+3x=4x+6$. Substitute $x = 8$ into the expression for $BD$.
$BD=4\times8+6=32 + 6=38$

Answer:

20