# Lesson 17

Lines in Triangles

## 17.1: Folding Altitudes (5 minutes)

### Warm-up

Students construct the altitudes of a triangle and observe that all 3 segments intersect at a single point.

### Student Facing

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

### Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

### Anticipated Misconceptions

If necessary, remind students that the definition of altitude is a line through a vertex perpendicular to the opposite side of the triangle.

If students aren’t sure how to fold an altitude, ask them how they would fold the perpendicular bisector of a side. Then, how can they modify that fold to create an altitude?

If students struggle to visualize what the altitudes will look like and therefore have trouble folding them, an index card can be a useful way to help “see” the 90 degree angle.

### Activity Synthesis

Ask students to look at triangles of students near them. What do they notice? (Everyone’s altitudes seem to intersect at one point.)

Tell students they’ll verify if this is true for *all* triangles in the next activity.

## 17.2: Altitude Attributes (15 minutes)

### Activity

In this activity students use the structure of the coordinate plane to examine the observation that the altitudes of a triangle all intersect at a single point. Students practice writing equations for perpendicular lines as they represent altitudes algebraically. Then they solve the system of equations (a fairly simple system with \(x=2\) as one equation) to show that the altitudes all intersect at a single point.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

### Launch

*Representation: Internalize Comprehension.*Activate or supply background knowledge about perpendicular bisectors. Display the following information: the perpendicular bisector of a segment is a line through the midpoint of the segment that is perpendicular to it. Provide a visual example to accompany this definition.

*Supports accessibility for: Memory; Conceptual processing*

### Student Facing

Triangle \(ABC\) is graphed.

- Find the slope of each side of the triangle.
- Find the slope of each altitude of the triangle.
- Sketch the altitudes. Label the point of intersection \(H\).
- Write equations for all 3 altitudes.
- Use the equations to find the coordinates of \(H\) and verify algebraically that the altitudes all intersect at \(H\).

### Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

### Student Facing

#### Are you ready for more?

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.

### Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Extension Student Response.

### Anticipated Misconceptions

If students aren’t sure how to find the slopes of the altitudes, ask them about the relationship between the slope of a side and the slope of the altitude through that side. (The product of the slopes is -1 since they are perpendicular.)

If students struggle to verify algebraically that \(H\) is the point of intersection, ask them what’s true if a particular point is on 3 different lines (the lines must intersect at that point, unless they coincide). How can students test if \(H\) is on each line?

As in the previous activity, an index card can be a useful too to help visualize the altitudes.

### Activity Synthesis

Ask students what the relationship is between the slope of a side and the slope of the altitude through that side. (The product of the slopes is -1 since they are perpendicular.) Invite students to share strategies for verifying their coordinates of \(H\). (Sample response: Start with \(x=2\), find the \(y\)-value using one equation, and test that point in the other equation.)

*Speaking: MLR8 Discussion Supports.*As students discuss the relationship between the slope of a side and the slope of the altitude through that side, press for details by asking students how they know that the product of the slopes is \(\text-1\). Also ask how they determined the equations for all three altitudes. Show concepts multi-modally by drawing and labeling a slope triangle for segment \(AC\) or \(BC\) and the altitude through that side. This will help students justify the slopes and equations for all three altitudes of the triangle.

*Design Principle(s): Support sense-making; Optimize output (for justification)*

## 17.3: Percolating on Perpendicular Bisectors (10 minutes)

### Optional activity

Students repeat the process from the warm-up and previous activity, this time studying perpendicular bisectors. They continue studying the same triangle so both the structure and some details (slopes) carry through to this activity.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

### Launch

Draw another triangle on tracing paper. Fold the perpendicular bisector of each side. What do you notice? (These segments also intersect at a single point.) Remind students that the definition of a perpendicular bisector is a line passing through the midpoint of a segment, perpendicular to that segment.

### Student Facing

Triangle \(ABC\) is graphed.

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

### Student Response

Teachers with a valid work email address can click here to register or sign in for free access to Student Response.

### Anticipated Misconceptions

Use the same slopes from the previous activity.

If students confuse altitudes, medians, and perpendicular bisectors, remind them that altitudes and medians must go through the triangle’s vertices, but the perpendicular bisectors don’t necessarily do so.

### Activity Synthesis

Invite students to share strategies for verifying their coordinates of \(P\). (Sample response: Start with \(x=4\), find the \(y\)-value using one equation, and test that point in the other equation.)

*Speaking: MLR8 Discussion Supports.*As students share their strategies for verifying the coordinates of \(P\), press for details by asking students how they determined the equations for all three perpendicular bisectors. Show concepts multi-modally by drawing and labeling the midpoint of segment \(AC\) and a slope triangle for the perpendicular bisector of \(AC\). This will help students justify the equations for all three perpendicular bisectors of the triangle.

*Design Principle(s): Support sense-making; Optimize output (for justification)*

## 17.4: Perks of Perpendicular Bisectors (10 minutes)

### Optional activity

Students further investigate the point of intersection of the perpendicular bisectors by observing and then proving the point is equidistant from the vertices. Students have the opportunity to practice writing the equation of a circle and then using that equation to reason about the vertices of a triangle. Monitor for students who verify by checking distances and students who verify by checking points in the equation.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

### Student Facing

Consider triangle \(ABC\) from an earlier activity.

- 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.
- Write the equation of a circle with center \(P\) and radius \(AP\).
- Construct the circle. What do you notice?
- Verify your hypothesis algebraically.

### Student Response

### Activity Synthesis

Invite the previously selected students to share their methods of verification. “What is the definition of a circle?” (The set of points equidistant from the center.) “Why do both of these methods work?” If no one used one of the methods, bring it up and ask students if it would work. (Points that work in the equation must have a distance of 5.9 from \(P\). Points that have a distance of 5.9 from \(P\) are on the circle by definition.)

*Speaking: MLR8 Discussion Supports.*As students share their strategies for verifying that points \(B\) and \(C\) are on the circle, press for details by asking students how they determined the equation of the circle. Show concepts multi-modally by drawing and labeling the center of the circle \(P\) and the radius \(AP\). This will help students justify the equation of the circle.

*Design Principle(s): Support sense-making; Optimize output (for justification)*

*Action and Expression: Develop Expression and Communication.*Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their ideas. For example: “I verified by . . . .”, and “I used the equation because . . . .”

*Supports accessibility for: Language; Organization*

## 17.5: Amazing Points (15 minutes)

### Optional activity

Students combine their work from this lesson and the previous lesson on triangle centers. They plot 3 centers for the same triangle (medians, altitudes, and perpendicular bisectors) and observe that the centers are collinear. When students are working on their proofs of this observation, monitor for those who write an equation for the line going through 2 of the points then substitute the third point into the equation, and for others who look at the slopes between the points.

### Launch

Reference the diagrams from the previous activities. The point of intersection of the altitudes was \(H=(2,1.2)\) and the point of intersection of the perpendicular bisectors was \(P=(4,4.4)\).

*Action and Expression: Develop Expression and Communication.*To help get students started, display sentence frames for the conjecture such as: “I notice . . . .”, and “The points are all. . . .”

*Supports accessibility for: Language; Organization*

### Student Facing

Consider triangle \(ABC\) from earlier activities.

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

### Student Response

### Activity Synthesis

Invite students to share their strategies for proving the 3 points are collinear. Then tell students, “This is called the Euler Line. This happens in *all *triangles, not just this one.”

*Speaking: MLR8 Discussion Supports.*As students share their strategies for proving the three points are collinear, press for details by asking students how they determined the equation of the line that goes through \(H\) and \(P\). Also, ask how they know that \(J\) is on the line. Show concepts multi-modally by drawing and labeling the points \(H\), \(P\), and \(J\) and the Euler line. This will help students justify why points \(H\), \(P\) and \(J\) are on the same line.

*Design Principle(s): Support sense-making; Optimize output (for justification)*

## 17.6: Tiling the (Coordinate) Plane (20 minutes)

### Optional activity

Students saw tessellations earlier in this course when they made Voronoi diagrams and then colored in the new tessellation created by the perpendicular bisectors. In this activity they will draw their own tessellation and practice writing equations of parallel, perpendicular, and intersecting lines.

### Launch

Tell students that tessellations are infinite but they need not spend the entire day drawing the first tessellation in this activity. Folding graph paper in half gives 4 sections to work in (front and back). Instruct students to consider half the page their “plane.”

If students use horizontal and vertical lines for the rectangles, tell them to make right triangles without using either horizontal or vertical lines.

*Representing, Conversing: MLR7 Compare and Connect.*Use this routine to prepare students for the whole-class discussion about the equations of the lines that outline a shape in their tessellation. After students create a tessellation for a shape of their choice, invite them to create a visual display of their work. Then ask students to quietly circulate and observe at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about their tessellations. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify the language students use to justify why lines in the tessellation are parallel or perpendicular.

*Design Principle(s): Cultivate conversation*

### Student Facing

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

- Tile the plane with congruent rectangles:
- Draw the rectangles on your grid.
- Write the equations for lines that outline 1 rectangle.

- Tile the plane with congruent right triangles:
- Draw the right triangles on your grid.
- Write the equations for lines that outline 1 right triangle.

- Tile the plane with any other shapes:
- Draw the shapes on your grid.
- Write the equations for lines that outline 1 of the shapes.

### Student Response

### Anticipated Misconceptions

If students struggle to find a third shape that tiles the plane, suggest they consider equilateral triangles or regular hexagons.

### Activity Synthesis

Invite several students to share their equations for the right triangle. Ask the class how they could check if these sets of equations outline right triangles. (Graph them or verify that a pair of slopes has a product of -1.)

*Representation: Develop Language and Symbols.*Use virtual or concrete manipulatives to connect symbols to concrete objects or values. Provide a variety of shapes, including some that do not tile the plane, for students to test and trace.

*Supports accessibility for: Visual-spatial processing; Conceptual processing*

## Lesson Synthesis

### Lesson Synthesis

Invite students to record some ideas about how coordinate proofs are different from other proofs. “Why did we save these conjectures for the coordinate proof unit?”

Students might discuss the precision of finding a coordinate pair and the algebraic nature of proof in this unit as different from the focus on congruence, angles, and triangles in the previous units. Students may comment that today’s conjectures were only examined for a particular case. Tell them that the ideas from this particular lesson are challenging to prove in general.

## Student Lesson Summary

### Student Facing

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 \(y=6\).