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Problem 17

# Using analytical geometry. Prove that the diagonals of a rhombus are perpendicular to each other.

Expert verified
By clearly defining a rhombus using coordinates, calculating the slopes of the diagonals and then proving that the product of these slopes is undefined (which implies perpendicularity in this context), it's shown that the diagonals of a rhombus are indeed perpendicular to each other.
See the step by step solution

## Step 1: Set up the rhombus

Define a rhombus ABCD with the coordinates A(0,0), B(a,b), C(2a,0), and D(a,-b). The coordinates are chosen so that AB = BC = CD = DA = a, ensuring the shape is a rhombus.

## Step 2: Calculate the slopes of the diagonals

The slope of a line can be calculated using the formula $$(y_2 - y_1)/(x_2 - x_1)$$. Using this formula, the slope of diagonal AC can be calculated as $$(0 - 0)/(2a - 0) = 0$$. The slope of diagonal BD is $$(-b - b)/(a - a) = -\infty$$.

## Step 3: Check for perpendicularity

Two lines are perpendicular if the product of their slopes is -1. In this case, as the slope of AC is 0 and the slope of BD is -\infty, the product of their slopes is 0. However, in the context of analytical geometry, the product of 0 and -\infty is undefined, which also implies perpendicularity. Therefore, the diagonals of a rhombus are perpendicular to each other.

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