Answer: 4 - It is not possible for the centroid and the orthocenter of a triangle to be the same point. The centroid is the point where the three medians of the triangle intersect, while the orthocenter is the point where the three altitudes of the triangle intersect. In general, these two points are not the same, although there are some special cases where they can coincide. For example, in an equilateral triangle, the centroid and the orthocenter are both the same point, which is the center of the triangle.
5. Let G be the centroid of triangle AXYZ. By definition, the medians XM and YN intersect at G, and they divide each other in a 2:1 ratio. This means that XK = 2GM, where GM is the length of the median from vertex A to midpoint M of YZ.
We can use the formula for the length of a median in a triangle to find GM. Let a be the length of side YZ, which is opposite to vertex A. Then, we have:
GM = sqrt((2b^2 + 2c^2 - a^2) / 4)
where b and c are the lengths of sides XZ and XY, respectively. Since we do not know the lengths of sides XZ and XY, we cannot directly compute GM.
However, we do know that the medians XM and YN intersect at K, which means that K is also the midpoint of XY. Therefore, we have:
b = XK = 21
c = YK = 2GM
Now, we can use the Pythagorean theorem to find a. Let H be the foot of the altitude from vertex A to YZ. Then, we have:
a^2 + H^2 = 4b^2/3
Since H is also the length of the altitude from vertex A to XM, we can use the formula for the length of an altitude to find H:
H = sqrt(b^2 - (GM/2)^2)
Substituting the values of b and GM, we get:
H = sqrt(441 - (c/2)^2)
Now, we can substitute the values of a and H into the equation above to get:
(2b^2 + 2c^2 - a^2) / 4 = a^2 + H^2
Simplifying and solving for c, we get:
c = sqrt((9b^2 - 4a^2 + 12H^2) / 4)
Substituting the values of a, b, and H, we get:
c = sqrt((189 - 4a^2 + 12(441 - (c/2)^2)) / 4)
Simplifying and solving for c, we get:
c = 18sqrt(13)
Therefore, XK = 2GM = 2c/3 = 12sqrt(13).
6. To find the orthocenter of triangle ABC, we need to find the intersection point of the altitudes. An altitude is a line segment that passes through a vertex of the triangle and is perpendicular to the opposite side.
We can start by finding the equations of the lines that contain the sides of the triangle. The line containing side AB has a slope of -1/2 (since it passes through points A(0,5) and B(10,0)) and passes through the midpoint of AB, which is (5, 2.5). Therefore, its equation is:
y - 2.5 = (-1/2)(x - 5)
Simplifying, we get:
y = -0.5x + 5.5
Explanation: