Question

The circumcentre and the orthocentre of the triangle formed by the sides \(\mathrm{y}=0, \mathrm{x}=0\) and \(2 \mathrm{x}+3 \mathrm{y}=6\) are respectively (1) \((3,2),(0,0)\) (2) \(\left(\frac{3}{2}, 1\right),(0,0)\) (3) \((-3,1),(0,0)\) (4) \(\left(-\frac{3}{2}, 1\right),(0,0)\)

Short answer

Answer: (1.5, 1) and (0, 0)

Step-by-step solution

Find the vertices of the triangle

To find the vertices, we will use the intersection points of the given lines: y=0, x=0, and 2x+3y=6. Intersection of y=0 and x=0: (0,0) Intersection of y=0 and 2x+3y=6: 2x+3(0)=6 => x=3 (3,0) Intersection of x=0 and 2x+3y=6: 0+3y=6 => y=2 (0,2) So, the vertices of the triangle are: A(0,0), B(3,0), and C(0,2).

Find the midpoints and slopes of the sides of the triangle

Next, we need to find the midpoints and slopes of each side to find the circumcentre. Midpoint of side AB [(3+0)/2, (0+0)/2] is D(1.5, 0). The slope of side AB is (0-0)/(3-0) = 0. Then, the slope of the perpendicular bisector of side AB is undefined. Midpoint of side AC [(0+0)/2, (0+2)/2] is E(0,1). The slope of side AC is (2-0)/(0-0) = undefined. Then, the slope of the perpendicular bisector of side AC is 0.

Find the circumcentre and orthocentre

Now we have the midpoints and slopes of the perpendicular bisectors of the sides AB and AC. We can find the intersection point of the lines DE and EF, which will be the circumcentre. Since the slope of DE is undefined and the slope of EF is 0, this means that line DE is a vertical line (x=1.5) and line EF is a horizontal line (y=1). Their intersection point is the circumcentre, which is (1.5, 1). To find the orthocentre, we need to see where the altitudes of the triangle intersect. Altitude from vertex A is a vertical line (x=0), and altitude from vertex B is a horizontal line (y=0). Their intersection point is the orthocentre, which is (0, 0). Thus, the circumcentre and the orthocentre of the triangle are (1.5, 1) and (0, 0), respectively. This corresponds to option (2).

Key concepts

Circumcentre

The circumcentre of a triangle is a critical point that can sometimes feel abstract. Think of it as the exact center of a circle that could wrap perfectly around the triangle, touching all three vertices. This 'circumscribed circle' is the ticket to understanding circumcentre.

To find this elusive point, one helpful visual is the construction of perpendicular bisectors for each side of the triangle. These bisectors, as the name suggests, cut each side in half at a 90-degree angle. Where they meet—that's your circumcentre! For a right-angled triangle, this gets even simpler; the midpoint of the hypotenuse serves as the circumcentre. This is because the hypotenuse effectively becomes the diameter of the circumscribed circle.

In our textbook exercise, the circumcentre was found by intersecting the perpendicular bisector of two sides. We don't always need all three, because two will suffice to find the exact point where they meet, determining the circumcentre's coordinates.

Orthocentre

Now, let's shift gears and tackle the orthocentre. This is another central point within a triangle but with a different flavor. The orthocentre is the spot where all the altitudes of a triangle, which are lines drawn from each vertex to the opposite side at a right angle, intersect. Here's a fun fact: the orthocentre's position inside the triangle can actually tell you the type of triangle you're dealing with.

For an acute triangle, the orthocentre sits comfortably inside; for a right triangle, it's right there at the vertex of the right angle; and for an obtuse triangle, it 'steps out' sitting outside the triangle. In the exercise, we dealt with a right triangle, and our orthocentre coincided with the right-angled vertex. It's quite straightforward since all lines drawn at right angles to the opposite sides from the right-angled vertex obviously intersect exactly at that vertex.

Triangle Vertices

Triangle vertices are the fundamental building blocks of any triangle—think of them as anchor points. To uncover these crucial points, you look for where the sides (or edges) of the triangle intersect. Each triangle has three vertices, which often get labeled with capital letters like A, B, and C in diagrams and exercises.

Knowing the vertices is paramount because they're the starting point for many triangle-related concepts, including finding the circumcentre and orthocentre mentioned earlier. In our exercise, by analyzing where the lines of the given equations intersected, we pegged down the vertices, laying the groundwork to solve for the circumcentre and orthocentre. It is the precise knowledge of these vertices that enables us to determine other significant properties and measures within the triangle.