Question

compare the distance formula and the pythagorean theorem by sketching a graph and showing work for both methods. either use colors or draw arrows between the different methods to show how they connect.

1. (9, -3) and (-1, 8)

distance formula \t\t\tgraph \t\t\tpythagorean theorem
graph of coordinate plane
find the distance between the points. round the answer to two decimal places.

1. (1,3) and (5,7) \t\t\t4. (-8, -9) and (-4, -10)
2. (10, 6) and (1, -4) \t\t\t6. (3, 2) and (8, 2)

Explanation:

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For Points (9, -3) and (-1, 8)

Step1: Define distance formula

The distance formula 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: Substitute values into formula

Let $(x_1,y_1)=(9,-3)$ and $(x_2,y_2)=(-1,8)$:
$$d = \sqrt{(-1-9)^2 + (8-(-3))^2} = \sqrt{(-10)^2 + (11)^2}$$

Step3: Calculate squared terms

$$d = \sqrt{100 + 121} = \sqrt{221} \approx 14.87$$

Step4: Pythagorean Theorem setup

Plot points, draw horizontal/vertical legs:
Horizontal change: $|-1-9|=10$, Vertical change: $|8-(-3)|=11$
Treat legs as $a=10$, $b=11$, distance = hypotenuse $c$:
$$c = \sqrt{a^2 + b^2} = \sqrt{10^2 + 11^2}$$

Step5: Solve for hypotenuse

$$c = \sqrt{100+121} = \sqrt{221} \approx 14.87$$

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For Points (1,3) and (5,7)

Step1: Apply distance formula

$$d = \sqrt{(5-1)^2 + (7-3)^2}$$

Step2: Compute and simplify

$$d = \sqrt{4^2 + 4^2} = \sqrt{16+16} = \sqrt{32} \approx 5.66$$

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For Points (-8,-9) and (-4,-10)

Step1: Apply distance formula

$$d = \sqrt{(-4-(-8))^2 + (-10-(-9))^2}$$

Step2: Compute and simplify

$$d = \sqrt{(4)^2 + (-1)^2} = \sqrt{16+1} = \sqrt{17} \approx 4.12$$

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For Points (10,6) and (1,-4)

Step1: Apply distance formula

$$d = \sqrt{(1-10)^2 + (-4-6)^2}$$

Step2: Compute and simplify

$$d = \sqrt{(-9)^2 + (-10)^2} = \sqrt{81+100} = \sqrt{181} \approx 13.45$$

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For Points (3,2) and (8,2)

Step1: Apply distance formula

$$d = \sqrt{(8-3)^2 + (2-2)^2}$$

Step2: Compute and simplify

$$d = \sqrt{5^2 + 0^2} = \sqrt{25} = 5.00$$

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Answer:

1. For (9, -3) and (-1, 8):

• Distance Formula Result: $\approx 14.87$
• Pythagorean Theorem Result: $\approx 14.87$
• Graph connection: The horizontal leg length = 10, vertical leg length = 11, forming a right triangle where the hypotenuse is the straight-line distance between the two points.

1. For (1,3) and (5,7): $\approx 5.66$
2. For (-8,-9) and (-4,-10): $\approx 4.12$
3. For (10,6) and (1,-4): $\approx 13.45$
4. For (3,2) and (8,2): $5.00$