Question

Coordinates of the vertices of a triangle ABC are (12,8), (-2,6) and (6,0) then the statement is -

• A. triangle is right but not isosceles
• B. triangle is isosceles but not right
• C. triangle is obtuse
• D. the product of the abscissa of the centroid, orthocenter and circumcenter is 160.

Answer 1

Answer: (D)

(D)

Answer 2

The triangle ABC is determined to be both right-angled at C and isosceles (AC=BC). We then calculated the abscissas of the centroid, orthocenter (vertex C), and circumcenter (midpoint of hypotenuse AB). The product of these abscissas was found to be 160.

1. Identify the vertices: Let A = (12,8), B = (-2,6), C = (6,0).

2. Calculate the square of the lengths of the sides: $AB^2 = (-2 - 12)^2 + (6 - 8)^2 = (-14)^2 + (-2)^2 = 196 + 4 = 200$ $BC^2 = (6 - (-2))^2 + (0 - 6)^2 = (8)^2 + (-6)^2 = 64 + 36 = 100$ $AC^2 = (6 - 12)^2 + (0 - 8)^2 = (-6)^2 + (-8)^2 = 36 + 64 = 100$

3. Determine the type of triangle: Since $BC^2 = AC^2 = 100$, the triangle is isosceles. Check for a right angle: $BC^2 + AC^2 = 100 + 100 = 200$. Since $AB^2 = 200$, we have $BC^2 + AC^2 = AB^2$. By the converse of the Pythagorean theorem, the triangle is a right-angled triangle with the right angle at vertex C. Thus, the triangle is both right-angled and isosceles.

4. Evaluate the given options: (A) triangle is right but not isosceles - False (it is isosceles). (B) triangle is isosceles but not right - False (it is right-angled). (C) triangle is obtuse - False (it is right-angled, not obtuse).

Let's check option (D): "the product of the abscissa of the centroid, orthocenter and circumcenter is 160."

5. Calculate the coordinates of the centroid (G), orthocenter (H), and circumcenter (O):

   • Centroid G: $G_x = \frac{12 + (-2) + 6}{3} = \frac{16}{3}$ $G_y = \frac{8 + 6 + 0}{3} = \frac{14}{3}$ So, G = (16/3, 14/3). Abscissa of G is $16/3$.

   • Orthocenter H: For a right-angled triangle, the orthocenter is the vertex where the right angle is located. The right angle is at C(6,0). So, H = (6,0). Abscissa of H is 6.

   • Circumcenter O: For a right-angled triangle, the circumcenter is the midpoint of the hypotenuse. The hypotenuse is AB. $O_x = \frac{12 + (-2)}{2} = \frac{10}{2} = 5$ $O_y = \frac{8 + 6}{2} = \frac{14}{2} = 7$ So, O = (5,7). Abscissa of O is 5.

6. Calculate the product of the abscissas: Product = $G_x \times H_x \times O_x = \frac{16}{3} \times 6 \times 5 = 16 \times 2 \times 5 = 16 \times 10 = 160$.