Triangle Centers: Centroid, Incentre, Orthocentre

What are the coordinates of the centroid of triangle ABC with vertices (x1, y1), (x2, y2), and (x3, y3)?

($rac{2x1 + x2 + x3}{4}, rac{2y1 + y2 + y3}{4}$)
($rac{x1 + x2 + x3}{2}, rac{y1 + y2 + y3}{2}$)
($rac{x1 + x2 + x3}{3}, rac{y1 + y2 + y3}{3}$) (correct)
($rac{x1 + 2x2 + 2x3}{5}, rac{y1 + 2y2 + 2y3}{5}$)

If the mid-point D of segment BC is given by the coordinates ($rac{x2 + x3}{2}, rac{y2 + y3}{2}$), what is the ratio that the centroid G divides AD?

2:3
1:1
2:1 (correct)
1:2

What are the coordinates of point G, the centroid of triangle ABC, expressed in terms of x1, x2, and x3?

($rac{x1 + x2 + x3}{3}, rac{y1 + y2 + y3}{3}$) (correct)
($rac{2x1 + x2 + x3}{4}, rac{y1 + 2y2 + y3}{4}$)
($rac{x1 + x2 + x3}{2}, rac{y1 + y2 + y3}{2}$)
($rac{x1 + x2 + x3}{3}, rac{y1 + 2y2 + y3}{5}$)

When considering the point D that bisects angle BAC, what geometric position does it represent?

The point where angle bisector AD intersects line segment BC (D) Signup and view all the answers

Given the coordinates of points A, B, and C of a triangle, how can the coordinates of the centroid G be calculated?

By averaging the x and y coordinates of the vertices (B) Signup and view all the answers

What is the relationship of the centroid to the orthocentre and circumcentre in terms of distance?

Centroid divides the line joining orthocentre and circumcentre in a 2:1 ratio. (B) Signup and view all the answers

In which type of triangle do the centroid, incentre, orthocentre, and circumcentre coincide?

Equilateral triangle (C) Signup and view all the answers

What does the nine-point circle touch?

Nine significant concyclic points of the triangle. (D) Signup and view all the answers

What is true about the relationship between the orthocentre, centroid, and circumcentre?

They are always collinear. (D) Signup and view all the answers

In an isosceles triangle, which points lie on the same line?

Centroid, orthocentre, incentre, and circumcentre. (C) Signup and view all the answers

How does the orthocentre relate to the right angle in a right-angled triangle?

It is located at the vertex where the right angle is formed. (C) Signup and view all the answers

What does HA represent in relation to the orthocentre?

Distance of orthocentre from vertex A. (A) Signup and view all the answers

What is the formula for the distance of the orthocentre from vertex A?

HA = 2R cos A (D) Signup and view all the answers

What is the definition of inclination in relation to a line intersecting the x-axis?

The measure of the smallest non-negative angle the line makes with the positive direction of the x-axis (D) Signup and view all the answers

What is the slope of a line if its inclination angle θ is defined as π/2?

Undefined (A) Signup and view all the answers

Which of the following correctly defines the slope of a non-vertical line?

The tangent of the inclination angle (C) Signup and view all the answers

If two points on a line are (x1, y1) and (x2, y2), how is the slope calculated?

m = (y2 - y1) / (x2 - x1) (B) Signup and view all the answers

For a line with a positive slope, which of the following statements is true regarding angle θ?

θ is greater than or equal to 0 degrees but less than 180 degrees (B) Signup and view all the answers

What is the slope of the x-axis?

0 (B) Signup and view all the answers

Which of the following describes the relationship when three points are collinear?

The slopes between each pair of points are equal (B) Signup and view all the answers

Given the slope formula m = tan(θ), what is the slope when θ is 0 degrees?

Both B and C are correct (A) Signup and view all the answers

What is the image of the point P(x1, y1) with respect to the x-axis?

(x1, -y1) (A) Signup and view all the answers

Given the equation of the line ax + by + c = 0, which condition indicates that the point P(x1, y1) is on the same side of the line as the origin?

If ax1 + by1 + c and c have the same sign (C) Signup and view all the answers

What is the formula used to find the image of point A(x1, y1) with respect to the line ax + by + c = 0?

B(h, k) = (-2(ax1 + bx1 + c)/a, -2(ax1 + bx1 + c)/b) (B) Signup and view all the answers

What is the correct equation of the line that is at a distance of 3 from the origin and makes an angle of 30° with the positive x-axis?

3x + y = 6 (B) Signup and view all the answers

If point P(x1, y1) is at (2, 3), what is its image with respect to the y-axis?

(-2, 3) (B) Signup and view all the answers

What is the image of the point P(x, y) with respect to the line y = x?

(y, x) (C) Signup and view all the answers

What distance does the point R1 divide the line joining P(x1, y1) and Q(x2, y2) in the ratio m:n represent?

The distance between P and Q (C) Signup and view all the answers

What does the formula $x ext{cos} heta + y ext{sin} heta = P$ determine in relation to lines?

The perpendicular distance from a point to a line (D) Signup and view all the answers

What is the equation of the bisector of the angle containing the origin if the lines are given by the equations with positive constant terms?

$\frac{ax + by + c}{a^2 + b^2} = + \frac{a'x + b'y + c'}{a'^2 + b'^2}$ (C) Signup and view all the answers

Given two lines intersecting with positive constants, when is the angle between them acute?

$a_1 a_2 + b_1 b_2 < 0$ (A) Signup and view all the answers

What determines if the angle bisector of two lines contains the origin or not?

The positivity of the constant terms c and c'. (C) Signup and view all the answers

For two lines defined by a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, how do you determine the acute angle bisector?

$\frac{a_1 x + b_1 y + c_1}{a_1^2 + b_1^2} = + \frac{a_2 x + b_2 y + c_2}{a_2^2 + b_2^2}$ (D) Signup and view all the answers

When comparing the acute angle bisector with the obtuse angle bisector, what condition indicates that the angle is obtuse?

$a_1 a_2 + b_1 b_2 > 0$ (D) Signup and view all the answers

What is the correct form of the bisector equation for the obtuse angle between two lines?

$\frac{a_1 x + b_1 y + c_1}{a_1^2 + b_1^2} = - \frac{a_2 x + b_2 y + c_2}{a_2^2 + b_2^2}$ (B) Signup and view all the answers

What adjustment is made to the equations of the lines to check if the bisectors contain the origin?

The constant terms must be made positive. (C) Signup and view all the answers