1.

1. Measure the lengths of the sides of the triangle PQR.
2. Which side of triangle ABC is the longest side?
3. How do you know which is the longest side of a right triangle without measuring it?

In a right triangle, the sides next to the right angle are called right angle sides. The side opposite the right angle is always the longest side. The triangle ABC will be written as ΔABC.

2.

1. Which sides are the right angle sides in ΔOAD?
2. Measure the length of AB in mm, and then find the area of the quadrangle ABCD.
3. Compute the area of the triangle OAD.
4. Jan-Hein said: “The area of the quadrangle ABCD is equal to the area of the square OEFG minus the area of the four right triangles.”
5. How long is the exact length of the side AD?

You can find the side of a square when you know the area by a reverse calculation. On your calculator, push the square root key.

3.

Annelotte’s house has a square garden that is 14 x 14 meters. In three corners of the garden are plants. One corner is tiled. The square in the center is for rabbits.

1. How many square meters is the whole garden?
2. How many square meters have been tiled?
3. What is the surface area of the area for the rabbits?
4. Annelotte has enclosed the rabbits’ square with wire mesh. How many meters of wire mesh did she use?

4.

5.

In photo 2, the second illustration above has been moved to partially cover the first illustration. The right angle sides of the yellow triangle are a and b, and the longest side of the yellow triangle is c.

1. Why is the area of the larger of the two squares equal to a²?
2. What is the area of the smaller of the two squares?
3. Explain why a² + b² = c².
4. Take a = 24 cm and b = 10 cm, and compute c.

In each right triangle, the sum of the areas of the two squares drawn on the right angle sides is equal to the area of the square drawn on the longest side. Call the lengths of the right angle sides of the triangle a and b, and the longest side c. Then a² + b² = c² holds. This is the theorem of Pythagoras. Pythagoras was a Greek who lived between 580 and 500 years before Christ.

6.

7.

Debby and Dik make tables for two different right triangles.

1. Copy the tables and fill them in.
2. Why did Dik write “square” above his table?

You can summarize the theorem of Pythagoras in a diagram. If you know the right angle sides of a right triangle, you can use this diagram to find the longest side. Here is how it looks. In the diagram, always list the longest side last.

8.

1. Draw a rectangle of 12 by 5 cm and draw a diagonal in the rectangle.
2. Calculate the length of the diagonal.
3. Measure your drawing to check whether your answer is correct. 9

9.

In triangle KLM, ∠K is 90°, KL = 15 cm and LM = 25 cm. Carmen makes the following table.

1. Draw triangle KLM and indicate which side is the longest.
2. Explain why Carmen’s table is incorrect.
3. Make a correct table and calculate side KM.

10.

1. In a right triangle, one right angle side is 9 cm and the longest side is 41 cm. Compute the length of the other right angle side.
2. In a right triangle the longest side is 65 meters and a right angle side is 60 meters. Compute the length of the other right angle side.

11.

In the following triangles, compute the length of the unknown side:

1. ∠B = 90°, AB = 6 and BC = 6
2. ∠D = 90°, DE = 20 and EF = 30
3. ∠P = 90°, QR = 25 and PQ = 8
4. ∠K = 90°, LM = 11 and KM = 4

12.

13.

14.

1. Draw a triangle PQR with vertices at P(1,1), Q(6,3), and R(3,5).
2. Compute the lengths of PQ, PR, and QR.
3. Show that triangle PQR is not a right triangle.

[Editorial note: This problem requires the converse of the Pythagorean theorem, which should be discussed in class either before this problem is assigned or in the discussion of the problem.]