Question

1. the coordinates of the vertices of the triangle are f(1,9), g(5,6), and h(5,9). what is length of segment fg in units? _______ units4. libby wants to measure the length of a pond. she measured 13 yd from point x to point z and 12 yd from point y to point z. what is the length of the pond?

Explanation:

Step1: Identify distance formula

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

Step2: Plug in F and G coordinates

For $F(1,9)$ and $G(5,6)$:
$$FG=\sqrt{(5-1)^2+(6-9)^2}$$

Step3: Calculate differences squared

Compute each term inside the square root:
$$FG=\sqrt{4^2+(-3)^2}=\sqrt{16+9}$$

Step4: Simplify to find length

Add and take the square root:
$$FG=\sqrt{25}=5$$

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Step1: Recognize right triangle setup

The pond length is the hypotenuse of a right triangle with legs 13 yd and 12 yd. Use the Pythagorean theorem: $a^2+b^2=c^2$, where $c$ is the pond length.

Step2: Substitute known values

$$c^2=13^2-12^2$$

Step3: Calculate squared terms

$$c^2=169-144=25$$

Step4: Solve for pond length

Take the square root:
$$c=\sqrt{25}=5$$

Answer:

1. 5 units
2. 5 yards