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De has endpoints d(–8,2) and e(9,8). Point f divides de into two parts with lengths in a ratio of 1:2. What are the two possible locations of f?

User Abacus
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Final Answer:

There are two possible locations for point f:

f(-2.33, 4)

f(3.33, 6)

Step-by-step explanation:

Ratio of segments: The problem states that point f divides segment de into two parts with a length ratio of 1:2. This means that the distance from d to f is one-third the total length of segment de, and the distance from f to e is two-thirds of the total length.

Calculate total length: First, find the total length of segment de using the distance formula:

de = sqrt((9 - (-8))^2 + (8 - 2)^2) = sqrt(17^2 + 6^2) = sqrt(305)

Divide by ratio: Divide the total length by 3 to find the distance from d to f:

df = de / 3 = sqrt(305) / 3 ≈ 5.88

Scale for second segment: Multiply df by 2 to find the distance from f to e:

fe = df * 2 = 5.88 * 2 ≈ 11.76

Parameterize line equation: We can represent the line segment de as the equation y = mx + b, where m and b are the slope and y-intercept, respectively. Since the endpoints are known, we can find the slope as:

m = (8 - 2) / (9 - (-8)) = 6/17

Use parameterization:

For the first location of f, plug df into the x-coordinate parameter and solve for the y-coordinate:

y = (6/17) * (-2.33) + b

Solve for b (y-intercept of point d) and obtain the first location: f(-2.33, 4)

Repeat the same process for the second location of f, using fe instead of df, to obtain: f(3.33, 6)

Therefore, the two possible locations for point f are f(-2.33, 4) and f(3.33, 6).

User CyberAleks
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