standard_description: (+)Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
problem_activity: 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.
[IMG]
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
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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
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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)


aligns: no