Question

If \(\triangle \mathrm{ABC}\) is a right angled triangle in which \(\mathrm{A}(3,0)\) and \(\mathrm{B}(0,5)\), then the coordinates of \(\mathrm{C}\) can be (1) \((5,3)\) (2) \((3,5)\) (3) \((0,0)\) (4) Both (2) and (3)

Short answer

Answer: The possible coordinates of vertex C are (5, 3) and (0, 0).

Step-by-step solution

Determine the slope of line AB

To form a right angle, the slope of the line connecting C to either A or B should be the negative reciprocal of the slope of line AB. First, determine the slope of the line AB using the coordinates of points A and B. Slope of line AB = \(\frac{y_B - y_A}{x_B - x_A}\) = \(\frac{5 - 0}{0 - 3}\) = \(-\frac{5}{3}\)

Find the slope of line AC and BC

Since the triangle is right-angled at vertex C, the slopes of lines AC and BC must be the negative reciprocals of the slope of line AB. Slope of line AC = \(-\frac{1}{-\frac{5}{3}}\) = \(\frac{3}{5}\) Slope of line BC = \(-\frac{1}{-\frac{5}{3}}\) = \(\frac{3}{5}\)

Apply point-slope form for lines AC and BC

Using point-slope form, we can write the equations for lines AC and BC using the slopes we just found and the coordinates of points A and B. Equation of line AC: \(y - y_A = m_{AC}(x - x_A)\) \(y - 0 = \frac{3}{5}(x - 3)\) Equation of line BC: \(y - y_B = m_{BC}(x - x_B)\) \(y - 5 = \frac{3}{5}(x - 0)\)

Solve for the coordinates of point C

Point C must lie on either line AC or BC. To find the coordinates of point C, we can plug the coordinates of each option into the equations of the lines and see which options fit the criteria for point C. Option 1: \((5, 3)\) For line AC: \(3 = \frac{3}{5}(5 - 3)\) becomes \(3 = \frac{6}{5}\). This is false. For line BC: \(3 = \frac{3}{5}(5)\) becomes \(3 = 3\). This is true. So, option 1 is a possibility. Option 2: \((3, 5)\) For line AC: \(5 = \frac{3}{5}(3 - 3)\). This is false. For line BC: \(5 = \frac{3}{5}(3)\). This is false. So, option 2 doesn't work. Option 3: \((0, 0)\) For line AC: \(0 = \frac{3}{5}(0 - 3)\). This is false. For line BC: \(0 = \frac{3}{5}(0)\). This is true. So, option 3 is also a possibility. Option 4: Both (2) and (3). This option is invalid since option 2 doesn't work. Therefore, the coordinates of point C can be either (5, 3) or (0, 0). The final answer is (1) \((5,3)\) and (3) \((0,0)\).

Key concepts

Slope of a Line

When we talk about the slope of a line, we're referring to a measure that describes how steep the line is. Essentially, the slope is the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line.

To calculate the slope between two points, labeled here as point A with coordinates \( (x_A, y_A) \) and point B with coordinates \( (x_B, y_B) \), you can use the formula:

• Slope (m) = \( \frac{y_B - y_A}{x_B - x_A} \).

For example, in the given problem, the slope of line AB with points A=(3,0) and B=(0,5) is computed as:

• \( \frac{5 - 0}{0 - 3} = -\frac{5}{3} \).

This indicates a downward slanting line from left to right, as denoted by the negative sign.

Point-Slope Form

The point-slope form is a specific way to write the equation of a line when you know a point on the line and the slope. It looks like this: \( y - y_1 = m(x - x_1) \).

This formula is handy because it directly incorporates the slope (m) and a known point, usually called the reference point (x₁, y₁). When you plug these values into the formula, it helps you develop the full equation of the line.

In our problem, we used points A and B to determine lines AC and BC. Here is how the equations were formed:

• For line AC, using point A (3,0) and slope \( \frac{3}{5} \): \( y - 0 = \frac{3}{5}(x - 3) \).
• For line BC, using point B (0,5) and slope \( \frac{3}{5} \): \( y - 5 = \frac{3}{5}(x - 0) \).

By using the point-slope form, it simplifies finding is the point C that lines up with these linear equations.

Negative Reciprocal

Understanding the concept of the negative reciprocal is crucial to problems involving right angles, especially involving slopes of perpendicular lines.
The negative reciprocal of a number is what you get when you flip the number (the reciprocal) and then change the sign.

Mathematically, if two lines are perpendicular, their slopes \( m_1 \) and \( m_2 \) satisfy \( m_1 \times m_2 = -1 \).

Consider the initial slope of line AB as \(-\frac{5}{3}\). The negative reciprocal is \( \frac{3}{5} \). This means any line with a slope of \( \frac{3}{5} \) and connected to either A or B will be perpendicular to AB.

• This is why AC and BC had their slopes as \( \frac{3}{5} \), ensuring they were perpendicular to AB at vertex C.

Understanding negative reciprocals helps ensure precision in identifying correct right angles in coordinate systems.