Lesson 17

Lines in Triangles

• Let’s investigate more special segments in triangles.

17.1: Folding Altitudes

Draw a triangle on tracing paper. Fold the altitude from each vertex.

17.2: Altitude Attributes

Triangle $$ABC$$ is graphed.

1. Find the slope of each side of the triangle.
2. Find the slope of each altitude of the triangle.
3. Sketch the altitudes. Label the point of intersection $$H$$.
4. Write equations for all 3 altitudes.
5. Use the equations to find the coordinates of $$H$$ and verify algebraically that the altitudes all intersect at $$H$$.

Any triangle $$ABC$$ can be translated, rotated, and dilated so that the image $$A’$$ lies on the origin, $$B’$$ lies on the point $$(1,0)$$, and $$C’$$ has position $$(a,b)$$. Use this as a starting point to prove that the altitudes of all triangles all meet at the same point.

17.3: Percolating on Perpendicular Bisectors

Triangle $$ABC$$ is graphed.

1. Find the midpoint of each side of the triangle.
2. Sketch the perpendicular bisectors, using an index card to help draw 90 degree angles. Label the intersection point $$P$$.
3. Write equations for all 3 perpendicular bisectors.
4. Use the equations to find the coordinates of $$P$$ and verify algebraically that the perpendicular bisectors all intersect at $$P$$.

17.4: Perks of Perpendicular Bisectors

Consider triangle $$ABC$$ from an earlier activity.

1. What is the distance from $$A$$ to $$P$$, the intersection point of the perpendicular bisectors of the triangle’s sides? Round to the nearest tenth.
2. Write the equation of a circle with center $$P$$ and radius $$AP$$.
3. Construct the circle. What do you notice?

17.5: Amazing Points

Consider triangle $$ABC$$ from earlier activities.

1. Plot point $$H$$, the intersection point of the altitudes.
2. Plot point $$P$$, the intersection point of the perpendicular bisectors.
3. Find the point where the 3 medians of the triangle intersect. Plot this point and label it $$J$$.
4. What seems to be true about points $$H, P,$$ and $$J$$? Prove that your observation is true.

17.6: Tiling the (Coordinate) Plane

A tessellation covers the entire plane with shapes that do not overlap or leave gaps.

1. Tile the plane with congruent rectangles:
1. Draw the rectangles on your grid.
2. Write the equations for lines that outline 1 rectangle.
2. Tile the plane with congruent right triangles:
1. Draw the right triangles on your grid.
2. Write the equations for lines that outline 1 right triangle.
3. Tile the plane with any other shapes:
1. Draw the shapes on your grid.
2. Write the equations for lines that outline 1 of the shapes.

Summary

The 3 medians of a triangle always intersect in 1 point. We can use coordinate geometry to show that the altitudes of a triangle intersect in 1 point, too. The 3 altitudes of triangle $$ABC$$ are shown here. They appear to intersect at the point $$(4,6)$$. By finding their equations, we can prove this is true.

The slopes of sides $$AB, BC,$$ and $$AC$$ are 0, $$\text-\frac{2}{3}$$, and 2. The altitude from $$C$$ is the vertical line $$x=4$$. The slope of the altitude from $$A$$ is $$\frac32$$. Since the altitude goes through $$(0,0),$$ its equation is $$y=\frac32 x$$. The slope of the altitude from $$B$$ is $$\text-\frac{1}{2}$$. Following this slope over to the $$y$$-axis we can see that the $$y$$-intercept is 8. So the equation for this altitude is $$y=\text-\frac{1}{2}x + 8$$.

We can now verify that $$(4,6)$$ lies on all 3 altitudes by showing that the point satisfies the 3 equations. By substitution we see that each equation is true when $$x=4$$ and $$y=6$$.