Activity 1 (van Hiele levels 0 and 1)



Activity 1 (van Hiele levels 0 and 1)

(A) The Pythagorean theorem.

The given GSP sketch shows a segment AB and a point P outside the segment.

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(1) Construct the line AP and the perpendicular from B to AP. Mark the intersection C.

(2) Hide the lines AC and BC, and replace them the segments AC and BC. [Do not hide the point P].

(3) Label the segments AB, AC, and BC by c, b, and a.

(4) Drag the point P to see the variation of the right triangle ABC.

(5) Mark the segments a, b, c. Measure their lengths by selecting Length from the Measure menu.

(6) From the Measure menu, select calculate (to see a calculator), and calculate

a ( a + b ( b – c ( c. [You input ``a’’ by clicking on the expression that shows the length of a, (, +, and – by clicking the calculator]. You should see that the answer for a ( a + b ( b – c ( c is 0.

(7) Drag the point P to change the right triangle ABC. Note that the lengths of a and b change, but not c. Note how the value of a ( a + b ( b – c ( c changes.

(8) Drag the point B (or A) to change the right triangle ABC. Note that the lengths of b and c change. Note how the value of a ( a + b ( b – c ( c changes.

You should note that this expression remains the same; it is always 0. This means that a*a + b*b = c*c.

This is the Pythagorean theorem: for a right triangle with sides a, b, c (in which c is the opposite side of the right angle), a ( a + b ( b – c ( c.

(B) a ( a + b ( b – c ( c when ABC is an obtuse triangle.

(1) Duplicate the page containing P and the right triangle. Mark the segment AC and select a point on it. Hide the segments AC, BC, and the point C. Label the new point you just selected C. In this way, you have a new triangle ABC.

(2) Measure angle ACB (by first marking the points A, C, B in this order, and then choose Angle from the Measure menu. You should note that this angle is always greater than 90 degrees. We call triangle ABC an obtuse triangle.

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(3) Label the segments AB, AC, and BC by c, b, and a. Measure these lengths and calculate a ( a + b ( b – c ( c.

(4) Drag the points P, C, B to vary the triangle. Note the variation of the value of the expression a ( a + b ( b – c ( c. What can you say about the sign of this value?

(C) a ( a + b ( b – c ( c when ABC is an acute triangle.

On a new page start with a segment AB.

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(1) Construct the perpendicular to AB at B. On this perpendicular select a point P.

(2) Join AP, and construct the perpendicular from B to AP. Mark the intersection point as Q.

(3) Hide all lines and construct the segment PQ.

(4) On the segment PQ select a point C.

(5) Hide the point Q, and construct the triangle ABC.

(6) Measure the angles of triangle ABC. You should note that they are all less than 90 degrees. Triangle ABC is called an acute triangle.

(7) Label the segments AB, AC, and BC by c, b, and a. Measure these lengths and calculate a ( a + b ( b – c ( c.

(8) Drag the points P, C, B to vary the triangle. Note the variation of the value of the expression a ( a + b ( b – c ( c. What can you say about the sign of this value?

(D) There is an alternative way to find a ( a. It is the area of the square constructed on the segment a.

(1) Return to the first page. Use a customized tool for the construction of a square to construct the squares on the segments a, b, and c.

(2) Mark the vertices of the square on a. Select interior from the construct menu. You will see a coloring and shading of the interior of the square. Select Area from the Measure menu. The area of the square will be recorded.

(3) Repeat the same for the squares on the segments b and c.

(4) Calculate Area of square on a + Area of square on b – Area of square on c. Drag the points P and B to see the variation of this sum.

Summary: In triangle ABC, if the lengths of the sides BC, CA, AB are a, b, c respectively, then the expression a ( a + b ( b – c ( c is zero, negative, or positive according as angle ACB is a right angle, an obtuse angle, or an acute angle.

Activity 2: Euclid’s proof of the Pythagorean theorem and its use in construction.

(A) Euclid’s proof of the Pythagorean theorem.

(1) Follow part (A) of Activity 1 to construct a right triangle ABC with a right angle at C.

(2) Use a customized tool to construct the squares ABDE, CAFG, and BCHK.

(3) Construct the perpendicular from C to AB to intersect DE at L and BC at M.

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To prove the Pythagorean theorem, Euclid showed that the area of the rectangle AMLE is the same as the area of the square CAFG. The same reasoning also showed that area of rectangle BMLD is the same as the area of the square BCHK. Since the square ABDE is made up of the two rectangles AMLE and BMLD, its area is the sum of the areas of the squares on AC and BC.

(4) Construct the interior of triangle ABF. How is its area related to the area of the square CAFG? Why?

(5) Construct the interior of triangle AEC. How is its area related to the area of the rectangle AMLE? Why?

(6) Give a simple explanation why the triangles ABF and AEC are congruent. You may find a very simple explanation by considering a rotation. But, also try to explain it using a congruence test.

(7) Now combine (4) to (6) above to prove the Pythagorean theorem.

(B) Converting a rectangle into a square of the same area.

The above proof of the Pythagorean theorem can be adapted to convert a given rectangle into a square of the same area. We begin with a rectangle AMEL, and proceed to construct the right triangle ABC. Then the square on AB has the same area as the given rectangle.

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(1) Construct a rectangle ABXY, say with the side AB longer than AY.

(2) Construct the midpoint O of AB, and the circle with AB as diameter.

(3) Construct a point M on AB such that AM = AY.

(4) Construct the perpendicular to AB at M, to intersect the circle in (2) at a point C.

(5) Construct triangle ABC. Why is angle ACB a right angle?

(6) Construct the square on CA. Why does this square have the same area as the given rectangle ABXY?

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