Find the slope of each side of the triangle.[br]
Find the slope of each altitude of the triangle.[br]
[list][*]Sketch the altitudes. [/*][*]Label the point of intersection [math]H[/math].[br][/*][/list][br]Write equations for all 3 altitudes.[br]
Use the equations to find the coordinates of [math]H[/math] and verify algebraically that the altitudes all intersect at [math]H[/math].[br]
[size=150]Any triangle [math]ABC[/math] can be translated, rotated, and dilated so that the image [math]A'[/math] lies on the origin, [math]B'[/math] lies on the point [math]\left(1,0\right)[/math], and [math]C'[/math] has position [math]\left(a,b\right)[/math].[/size][br][br]Use this as a starting point to prove that the altitudes of all triangles all meet at the same point.
Find the midpoint of each side of the triangle.[br]
[list][*]Sketch the perpendicular bisectors, using the Midpoint and Perpendicular Line tools. [/*][*]Label the intersection point [math]P[/math].[br][/*][/list]Write equations for all 3 perpendicular bisectors.[br][br]
Use the equations to find the coordinates of [math]P[/math] and verify algebraically that the perpendicular bisectors all intersect at [math]P[/math].[br]
What is the distance from [math]A[/math] to [math]P[/math], the intersection point of the perpendicular bisectors of the triangle’s sides? Round to the nearest tenth.[br]
Write the equation of a circle with center [math]P[/math] and radius [math]AP[/math].[br]
[list][*]Construct the circle. [/*][/list][br]What do you notice?[br]
Verify your hypothesis algebraically.[br]
[list][*]Plot point [math]H[/math], the intersection point of the altitudes.[br][/*][*]Plot point [math]P[/math], the intersection point of the perpendicular bisectors.[br][/*][*]Find the point where the 3 medians of the triangle intersect. Plot this point and label it [math]J[/math].[br][/*][/list][br]What seems to be true about points [math]H[/math], [math]P[/math], and [math]J[/math]?
Prove that your observation is true.[br]
[size=150]Tile the plane with congruent rectangles:[/size][br][br]Draw the rectangles on your grid.[br]
Write the equations for lines that outline 1 rectangle.[br]
[size=150]Tile the plane with congruent right triangles:[/size][br][br]Draw the right triangles on your grid.[br]
Write the equations for lines that outline 1 right triangle.[br]
[size=150]Tile the plane with any other shapes:[/size][br][br]Draw the shapes on your grid.[br]
Write the equations for lines that outline 1 of the shapes.[br]