Question

The circumcentre of the triangle formed by the lines \(\mathrm{x}+\mathrm{y}=0, \mathrm{x}-\mathrm{y}=0\) and \(\mathrm{x}-7=0\) is \(\ldots \ldots\) (a) \((7,0)\) (b) \((3.5,0)\) (c) \((0,7)\) (d) \((3.5,3.5)\)

Short answer

The circumcentre of the triangle formed by the lines $\mathrm{x}+\mathrm{y}=0, \mathrm{x}-\mathrm{y}=0$ and $\mathrm{x}-7=0$ is $(3.5, -3.5)$.

Step-by-step solution

Find the vertices of the triangle

To find the vertices, we need to solve the given linear equations in pairs: 1. x + y = 0 and x - y = 0 2. x + y = 0 and x - 7 = 0 3. x - y = 0 and x - 7 = 0 Solving the first pair of equations, we get: x = 0 and y = 0, which give us the first vertex A(0, 0). Solving the second pair of equations, we get: x = 7 and y = -7, which give us the second vertex B(7, -7). Solving the third pair of equations, we get: x = 7 and y = 7, which give us the third vertex C(7, 7).

Calculate the midpoints and slopes of two sides

Consider two sides AB and AC. Find the midpoints of these two sides: Midpoint of AB, M1 = \(\left(\frac{A_x + B_x}{2}, \frac{A_y + B_y}{2}\right)\) = \(\left(\frac{0+7}{2}, \frac{0-7}{2}\right)\) = \((3.5, -3.5)\) Midpoint of AC, M2 = \(\left(\frac{A_x + C_x}{2}, \frac{A_y + C_y}{2}\right)\) = \(\left(\frac{0+7}{2}, \frac{0+7}{2}\right)\) = \((3.5, 3.5)\) Find the slopes of the sides AB and AC: Slope of AB, m1 = \(\frac{B_y - A_y}{B_x - A_x}\) = \(\frac{-7-0}{7-0}\) = \(-1\) Slope of AC, m2 = \(\frac{C_y - A_y}{C_x - A_x}\) = \(\frac{7-0}{7-0}\) = \(1\)

Find the perpendicular bisectors of the two chosen sides

The slope of the line perpendicular to AB is the negative reciprocal of m1: m3 = \(\frac{1}{1}\) = \(1\). The slope of the line perpendicular to AC is the negative reciprocal of m2: m4 = \(\frac{1}{-1}\) = \(-1\). Now, we have the slopes and midpoints of the perpendicular bisectors M1R1 and M2R: Equation of M1R1: y - M1_y = m3 (x - M1_x) ⇒ y - (-3.5) = 1(x - 3.5) ⇒ y + 3.5 = x - 3.5 Equation of M2R: y - M2_y = m4 (x - M2_x) ⇒ y - 3.5 = -1(x - 3.5) ⇒ y - 3.5 = -x + 3.5

Calculate the intersection point of the two perpendicular bisectors

To find the circumcentre, we need to solve the two equations obtained in Step 3: 1. y + 3.5 = x - 3.5 2. y - 3.5 = -x + 3.5 Add both equations: 2y = -7 y = -3.5 Substitute the value of y in the first equation: -3.5 + 3.5 = x - 3.5 0 = x - 3.5 x = 3.5 Hence, the circumcentre is (3.5, -3.5), which is not among the given options.

Key concepts

Triangle Vertices

Understanding triangle vertices is crucial when identifying or analyzing the shape of a triangle. A vertex is simply a point where two or more lines meet, forming a corner. In the given exercise, the triangle is formed by three lines defined by the equations:

• x + y = 0
• x - y = 0
• x - 7 = 0

By solving these equations in pairs, we can determine where each pair intersects, revealing the triangle's vertices:

• The intersection of x + y = 0 and x - y = 0 yields the vertex A(0, 0).
• The intersection of x + y = 0 and x - 7 = 0 gives the vertex B(7, -7).
• The intersection of x - y = 0 and x - 7 = 0 results in the vertex C(7, 7).

Identifying these vertices is the first crucial step in understanding the geometry of the triangle.

Perpendicular Bisectors

Perpendicular bisectors are lines that perpendicularly cross another line segment at its midpoint. This concept is especially useful when finding a triangle's circumcenter, as the circumcenter is the point where the perpendicular bisectors of a triangle's sides intersect.
After identifying the triangle's vertices, we move to the calculation of its perpendicular bisectors.
For a line segment like AB or AC, the slope of the perpendicular bisector is the negative reciprocal of the original slope. For example:

• The slope of AB is -1; thus, the perpendicular slope becomes 1.
• The slope of AC is 1; thus, its perpendicular slope becomes -1.

With these perpendicular slopes in hand, we can construct the equations for these bisectors, using the midpoints we found earlier to find the triangle's circumcenter.

Midpoints of Sides

Midpoints play a vital role in understanding the properties of triangles, particularly when looking for the circumcenter. A midpoint is the point exactly halfway along a side of the triangle. To find it, you calculate the average of the coordinates of the endpoints.
In our example, we need these midpoints to determine the bisectors:

• The midpoint of side AB is calculated as follows: \( \left( \frac{0+7}{2}, \frac{0-7}{2} \right) \), resulting in (3.5, -3.5).
• Similarly, the midpoint of side AC is: \( \left( \frac{0+7}{2}, \frac{0+7}{2} \right) \), which is (3.5, 3.5).

These midpoints are critical as they help define the location for constructing the perpendicular bisectors, bringing us one step closer to locating the circumcenter.

Slopes of Lines

The slope of a line is a measure of its steepness and direction. In the context of triangles, understanding line slopes is essential for finding perpendicular bisectors. Slopes help determine how lines relate to one another:

• Lines with slopes that are negative reciprocals are perpendicular to each other.

When dealing with a triangle, these relationships are essential for defining perpendicular bisectors which converge at the circumcenter.

• For side AB, with coordinates (0,0) and (7,-7), the slope is calculated by \( \frac{-7}{7} = -1 \).
• For side AC with coordinates (0,0) and (7,7), the slope is \( \frac{7}{7} = 1 \).

Using these slopes, we determine the perpendicular slopes required to find the bisectors necessary for locating the triangle's circumcenter. Understanding how to compute and apply these slopes is a key skill in geometry.