Given: BE and AD are altitudes (intersecting at H) of triangle ABC, while F, G, and K are midpoints of AH, AB and BC, respectively. Prove: angle FGK is a right angle.

Solution :

Since is an altitude of a triangle , it is perpendicular to (since the triangle ).

Since and are the midpoints, then the line and the line . The line must be parallel to the line and is therefore perpendicular to

Now since F and are the mid points, we get  . The line is parallel to BH  and is perpendicular to AC.

Therefore, we have

parallel to

perpendicular to

parallel to

Thus is perpendicular to and also ∠ is a right angle.

when rotating a figure 180 degrees, imagine that you are able to take the figure and turn it clockwise or counterclockwise around a center point. to rotate a figure 180 degrees, you will need to apply the rule (x, y) → (-x, -y). start by using a coordinate grid with coordinates for each vertex of the figure. the center point of the coordinate grid is located at (0, 0), which is what you will rotate the figure around. write down the original coordinates of the shape you are going to rotate. then, apply the rule. for example, coordinate (1, 2) becomes (-1, -2). another example would be, coordinate (-4, -2) becoming (4, 2). after you change each original coordinate to the rotated coordinate, you will draw your new figure.

step-by-step explanation: