Question

1. sandy wants to place two rocking chairs and a sofa on her back porch in a way that allows guests to be able to see each other while also enjoying her pretty flowers in the backyard. on a coordinate grid, one rocking chair is at (0, 2), another at (3, 6), and the sofa at (7, 1).

determine how far apart people are when sitting in the two rocking chairs and on the sofa.

a metal support frame forms triangle ghi with coordinates g(2, 4), h(6, -2), and i(10, 4).
slope of gh: ________
slope of hi: ________
triangle ghi is a(n):
a. right triangle
b. acute triangle
c. obtuse triangle

Explanation:

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First Problem (Distances between furniture)

Step1: Define distance formula

The distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$.

Step2: Distance between rocking chairs

Points $(0,2)$ and $(3,6)$:

$$\begin{align*} d_1&=\sqrt{(3-0)^2+(6-2)^2}\\ &=\sqrt{3^2+4^2}\\ &=\sqrt{9+16}\\ &=\sqrt{25}=5 \end{align*}$$

Step3: Distance, chair to sofa (0,2) to (7,1)

$$\begin{align*} d_2&=\sqrt{(7-0)^2+(1-2)^2}\\ &=\sqrt{7^2+(-1)^2}\\ &=\sqrt{49+1}\\ &=\sqrt{50}=5\sqrt{2}\approx7.07 \end{align*}$$

Step4: Distance, chair to sofa (3,6) to (7,1)

$$\begin{align*} d_3&=\sqrt{(7-3)^2+(1-6)^2}\\ &=\sqrt{4^2+(-5)^2}\\ &=\sqrt{16+25}\\ &=\sqrt{41}\approx6.40 \end{align*}$$

Step1: Slope formula definition

Slope between $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2-y_1}{x_2-x_1}$.

Step2: Calculate slope of GH

Points G(2,4) and H(6,-2):

$$\begin{align*} m_{GH}&=\frac{-2-4}{6-2}\\ &=\frac{-6}{4}=-\frac{3}{2} \end{align*}$$

Step3: Calculate slope of HI

Points H(6,-2) and I(10,4):

$$\begin{align*} m_{HI}&=\frac{4-(-2)}{10-6}\\ &=\frac{6}{4}=\frac{3}{2} \end{align*}$$

Step4: Verify triangle type

Check slopes: $m_{GH} \times m_{HI} = -\frac{3}{2} \times \frac{3}{2}=-\frac{9}{4}
eq -1$, so no right angle. Now use the law of cosines. First find all side lengths:

• $GH=\sqrt{(6-2)^2+(-2-4)^2}=\sqrt{16+36}=\sqrt{52}$
• $HI=\sqrt{(10-6)^2+(4+2)^2}=\sqrt{16+36}=\sqrt{52}$
• $GI=\sqrt{(10-2)^2+(4-4)^2}=\sqrt{64+0}=8$

Law of cosines for angle at I: $GH^2 = GI^2 + HI^2 - 2(GI)(HI)\cos\theta$

$$\begin{align*} 52&=64+52-2(8)(\sqrt{52})\cos\theta\\ 52&=116-16\sqrt{52}\cos\theta\\ -64&=-16\sqrt{52}\cos\theta\\ \cos\theta&=\frac{64}{16\sqrt{52}}=\frac{4}{\sqrt{52}}\approx0.5547>0 \end{align*}$$

All angles have positive cosine values, so all angles are acute.

Answer:

• Distance between the two rocking chairs: $5$ units
• Distance between (0,2) and the sofa: $5\sqrt{2}$ (or ~7.07) units
• Distance between (3,6) and the sofa: $\sqrt{41}$ (or ~6.40) units

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Second Problem (Triangle GHI)