Download Right Triangle with an Altitude

Right Triangle with an Altitude
What we’re doing: Looking at the relationships of the triangles formed by an altitude from the
right angle.
What you’ll need: An index card; a straight edge; a pen/pencil; a scissors (share?)
1. Use a straight edge to draw a diagonal on your index card. Do you know what a diagonal
is?
a. Cut the diagonal to get two triangles.
b. What kind of triangles do you have? How are they related to each other?
2. Draw an altitude from the right angle on both sides of both triangles.
-how can you be absolutely sure you have an altitude? This requires accuracy, if you
aren’t sure, check.
3. You have two congruent right triangles with altitudes.
a. Label the vertices of the triangle A, B, and C with the right angle B, the vertex from
the longer side C, and the vertex from the shorter side A. (of both triangles)
b. Label the point where the altitude intersects the hypotenuse D (on both sides of the
altitude).
c. Flip the triangles over and label the backs with the proper letters.
4. Cut ONE of the altitudes.
5. Rearrange the triangles so the two small triangles cover the larger triangle.
a. Notice that what we have is a right triangle with an altitude dividing it into two
smaller triangles.
b. Also notice that the A lines up on top of the A, the B on the B, the C on the C, and the
D on the D. Look at this as one triangle that can be split into parts creating 3
triangles.
6. Rearrange the triangles so they are stacked big-medium-small with both legs flush (right
angles on top of each other). This is just to emphasize that you have three right angles.
7. Shift the triangles so that the hypotenuse and long legs are flush (smallest angle stacked)
a. What can you say about the smallest angle of the triangles?
b. With all three triangles having right angles, and knowing this about the smallest
angles, what can you say about the triangles?
c. By what theorem?
8. Looking at your large triangle (with the two others drawn on it), use the labels to label the
vertices of the similar triangles below so corresponding parts line up.
9. Lay out your cut out triangles in this fashion. Check the labeled points on your cut out
triangles with the labeled triangles above. Do they match up? If not, find out what’s wrong.
Use the labeled figures to answer the questions below. You may wand to use descriptions such as
small triangle, medium triangle, large triangle; and short leg, long leg, hypotenuse.
1. What is the long leg of the small triangle?
2. List the hypotenuse of all three triangles.
3. How many times does segment BD appear in the triangles? Describe where you found
them.
4. What two sides correspond with BA of the big triangle?
5. If CB is 24 and CA is 36, set up a proportion to find CD.
6. Use your results from number 5 to find BA and DB. How are you going to find these (it’s
not by a proportion)?
7. Use the results to number six to find DA.
8. Label the lengths of all sides of the triangles above.
9. What two formulas/problem solving techniques will you be using in this section?