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Exam 4 Exercises

Problem 1)

Consider the three altitudes that a right triangle has. How many of them lie inside the triangle? How many lie on the triangle? How many lie outside the triangle? If the triangle is acute? If the triangle is obtuse?

Problem 2)

Is it possible for the medians of a triangle to lie outside the triangle?

Problem 3)

If an isosceles triangle has an angle of 80 degrees, another angle in the triangle could have measure I. 80 degrees II. 100 degrees III. 50 degrees a) I only b) I, II, and III c) I and III only d) I and II only

Problem 4)

Consider a vertex of a triangle and the altitude, median, and angle bisector drawn through it. Under what condition will these three lines be the same line?

Problem 5)

In the diagram below, triangles and are drawn sharing an edge. If it is given that then which sides would you need to know are congruent, in order to infer that the triangles are congruent? If you were instead given that some other angle were congruent, rather than a side, would it matter which angle is given?

Problem 6)

What is the angle measure of any angle in an equilateral triangle?

Problem 7)

In an equilateral triangle , choose vertex B and draw the median from that vertex. Label the point where the median intersects the opposite side as D. Find the interior angle measures of .

Problem 8)

In the figure below the triangle is isosceles. The angle at vertex C is 32 degrees. The line drawn at vertex B is the angle bisector. Find all angles in the figure. When you're done, look at all of the angles and see if you can make an interesting observation about the triangles in the picture. Here is a very difficult problem, but one that is similar to the work we've done on right triangles: Suppose that the length of AB is 1. What is the length of BC? This is a famous historical problem, with a pretty interesting answer. If you can't figure this out in about five or ten minutes, don't worry about it. We can talk about it when I see you next time.

Problem 9)

Suppose is equilateral and edge is extended to a point D so that . Find all of the angles in .

Problem 10)

In the figure below and . If and and and find the values of x and y.

Problem 11)

An astronomer sees a planet traveling in what appears to be a circular orbit. However, no star is visible and she hypothesizes that the planet is actually orbiting a black hole, or possibly a dense collection of dark matter. She would like to know where is the object that the planet is orbiting. Right now she observes the planet at point A and a little while later she observes it at B. Notice that this is inadequate information to figure out the path of motion of the planet, because many circles can be drawn to contain two points, like so.
However, when she makes a third observation, only one circle is now possible.
Describe how the astronomer could use the three points to find the center and radius of the circle.

Problem 12)

Prove that the diagonals of rhobmi are perpendicular bisectors.

Problem 13)

Let ABCD be an isosceles trapezoid with AD = BC and . Prove that .