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Problem 12:

Find the equation of one of the sides of an isosceles right-angled triangle whose hypotenuse is given by $3x + 4y = 4$ and the opposite vertex of the hypotenuse is $(2, 2)$.

We are given:

• The equation of the hypotenuse: $3x + 4y = 4$
• The opposite vertex of the hypotenuse: $(2, 2)$

An isosceles right-angled triangle has two sides perpendicular to each other, meeting at the right angle. Since the vertex opposite to the hypotenuse is $(2, 2)$, one of the sides will be perpendicular to the hypotenuse.

First, find the slope of the hypotenuse. For a line in the form $Ax + By = C$, the slope is $-\frac{A}{B}$.

Thus, the slope of $3x + 4y = 4$ is: $\text{slope of hypotenuse} = -\frac{3}{4}$

Since one of the sides is perpendicular to the hypotenuse, the slope of this side will be the negative reciprocal of $-\frac{3}{4}$, which is: $\text{slope of side} = \frac{4}{3}$

Now, use the point-slope form of the equation of a line. The point-slope form is: $y - y_1 = m(x - x_1)$ where $m$ is the slope, and $(x_1, y_1)$ is a point on the line. We are given the point $(2, 2)$, so the equation becomes: $y - 2 = \frac{4}{3}(x - 2)$ Simplify this equation: $y - 2 = \frac{4}{3}x - \frac{8}{3}$ Multiply through by 3 to clear the fraction: $3y - 6 = 4x - 8$ Rearrange to get the equation of the side: $4x - 3y = 2$

Thus, the equation of one side of the triangle is: $4x - 3y = 2$

Would you like to explore further solutions, or do you have any specific question on this step?

Here are some relative questions for more practice:

1. How would you find the other side of the isosceles triangle?
2. What is the length of the hypotenuse using the distance formula between the given vertices?
3. How can you verify that the sides form a right angle?
4. How do you determine if two lines are perpendicular based on their slopes?
5. How do you calculate the area of an isosceles triangle given its vertices?

Tip: When working with perpendicular lines, the product of their slopes should always equal $-1$.