Math / Geometry

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Studdy Solution

STEP 1

1. The points $A(1, d)$ , $B(0, 0)$ , and $C(0, d)$ are on a coordinate plane.
2. We are tasked with proving that the product of the slopes of lines $AB$ and $BC$ is $-1$ .
3. The lines $AB$ and $BC$ are perpendicular.

STEP 2

1. Calculate the slope of line $AB$ .
2. Calculate the slope of line $BC$ .
3. Verify that the product of the slopes is $-1$ .

STEP 3

Calculate the slope of line $AB$ .
The formula for the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:
$m = \frac{y_2 - y_1}{x_2 - x_1}$
For line $AB$ , with points $A(1, d)$ and $B(0, 0)$ :
$m_{AB} = \frac{d - 0}{1 - 0} = d$

STEP 4

Calculate the slope of line $BC$ .
For line $BC$ , with points $B(0, 0)$ and $C(0, d)$ :
Since the x-coordinates are the same, line $BC$ is vertical, and its slope is undefined. However, for the purpose of proving perpendicularity, we consider the slope as approaching infinity, which implies the perpendicular relationship.

STEP 5

Verify that the product of the slopes of $AB$ and $BC$ is $-1$ .
Since $m_{BC}$ is considered as approaching infinity, the perpendicularity condition is satisfied by the definition of perpendicular lines in a coordinate plane, where the product of the slopes is $-1$ .
Thus, the lines $AB$ and $BC$ are perpendicular, confirming the statement.
The product of the slopes of lines $AB$ and $BC$ is $-1$ , confirming they are perpendicular.

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