Question

1. ABCD is a rectangle whose side AB has the equation 3x - 12y +5 = 0. Find the slopes of the other
minutes
sides of the rectangle.
he equations x + 2 = 0 and y-3 = 0. The centra​

Answer 1

Answer:

When x = 0, y = 12/5. This is where the line AB intercepts the y-axis.

When y = 0, x = -4. This is where the line AB intercepts the x-axis.

Draw an (x, y)-coordinate system’s axes, and mark these two intercepts on it. Label the first point (on the y-axis) A and the second point (on the x-axis) B.

The slope of AB is given by the definition:

slope = rise / run

Or, if you draw the triangle OAB, where O is the origin of the (x, y)-coordinate system:

slope of AB = OA / BO = (12 / 5) / 4 = 3 / 5.

Because the line DA is at right angles to AB, its slope is the negative reciprocal of AB’s. That is,

For any two orthogonal lines, with slopes s1 and s2, neither of which is zero:

s1 . s2 = -1

Using this equation, we get that:

slope of DA = (-1) / (3 / 5) = -5 / 3.

Check: That’s what the formula tells us. But how do we know we can trust it? To make sure we don’t go astray by blindly following a formula, it would be smart to check that the answer we got is reasonable.

You can check this answer by constructing one of the possible rectangles ABCD (there’s an infinite number of them!) You’ll see that we are given no facts about the length of either BC or DA. That means we can choose to place C on the positive x-axis, where y = 0. If you’ve made your drawing carefully to scale, e.g. using squared paper or a grid in a vector drawing program, you’ll see that C falls at x = 36/25, as it should since it’s on the line AD, whose equation is:

slope = (change in y) / (change in x) = (y - y1) / (x - x1)

where y1 and x1 are the intercepts of the line AD with the two axes, so that y1 = 12/5.

To get the equation of AD, we need to use Pythagoras’ Theorem thrice:

AB^2 = OA^2 + OB^2 = (12/5)^2 + 4^2

AD^2 = OA^2 + OD^2 = (12/5)^2 + x1^2

BD^2 = AB^2 + AD^2

So (4 + x1)^2 = (12/5)^2 + 4^2 + (12/5)^2 + x1^2

Expanding the left-hand side of this equation, and cancelling like terms, we have:

x1 = 36 / 25

and we can apply the definition of a slope to show that this gives the right slope between points A and D on the two axes:

slope = rise / run = (12 / 5) / (-36 / 25) = -5 / 3.

Step-by-step explanation:

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