1.2 Why all the fuss about pythagoras?  (Page 2/3)

Substitution:AH ² + EH ² = (6) ² + (8) ² = 36 + 64 = 100 cm ² If E ² = 100 cm ² , then AE must be 10 cm.

2.1 Calculate the length of the third side of these triangles:

2.1.1 ΔDEF with D a right angle and e = 5 mm and f = 12 mm

2.1.2 ΔXYZ with Y a right angle and x = 3 cm and y = 5 cm.

3 Problem : What is the length of the shortest side (b) of the right–angled ΔABC if the other two sides are 6 cm and 9 cm? C is the right angle.

Solution : In a right–angled triangle the longest side is always the hypotenuse: the side opposite the right angle. Now we use the Theorem of Pythagoras in the other form.

• If b is the shortest side, and C is a right angle, then c must be the longest side. So, you use:

b ² = c ² – a ² (note carefully where b ² is, and that we subtract)

b ² = (9) ² – (6) ² = 81 – 36 = 45 cm ² Calculator time!

b ² = 45. Use the $\sqrt{}$ – button on the calculator to find the value of b.

• Your calculator supplies the answer: b = 6,7082039 . . . et cetera. But is this a sensible answer? Discuss whether the approximated (rounded) answer of 6,7cm is good enough for our purposes.

3.1.1 Calculate the length of the hypotenuse of a triangle with both of the other sides equal to 9 cm. (Label the triangle yourself.)

3.1.2 ΔPQR is right–angled and isosceles. Calculate the length of PR, if the hypotenuse is 13,5 cm.

4 Problem : Is ΔGHK right–angled if GK = 24 cm, GH = 26 cm and HK = 10 cm?

Solution : In this problem we know all three the sides’ lengths. If we want to find out whether it is right–angled, we have to confirm whether (hypotenuse) ² = (one side) ² + (other side) ² .

The hypotenuse is always the longest side. We have a very specific method whenever we have to confirm a result. We calculate the left–hand side and the right–hand side of the equation separately. Thus:

• Left–hand side = (hypotenuse) ² = 26 ² = 676 cm ²
• Right–hand side = (one side) ² + (other side) ² = 24 ² + 10 ² = 576 + 100 = 676 cm ²
• Because the left–hand side and right–hand side come out the same, we can conclude that the triangle is right–angled.

• Is it possible to know which angle is the right angle? You give the answer!

4.1 Are the triangles with the given side lengths right–angled? Which angle is the right angle?

4.1.1 a = 30 mm, b = 40 mm and c = 50 mm.

4.1.2 p = 8 cm, q = 13 cm and r = 15 cm.

4.1.3 MN = 15,56 cm, and NP = MP = 11 cm.

end of CLASS WORK

HOMEWORK ASSIGNMENT

1 Find the third side of the following triangles:

1.1 ΔABC with C = 90° and b = 5 mm and c = 13 mm

1.2 ΔMNO with O the right angle and m = 6 cm and n = 8 cm.

2 Determine whether the following triangles are right–angled, and which angle is 90° .

2.1 a = 9 mm, b = 11 mm and c 13 cm

2.2 XZ = 85 mm, XY = 13 mm and YX = 86 mm.

end of HOMEWORK ASSIGNMENT

The connection between roots and exponents

CLASS WORK

1 Eight of the equations in this list must be filled in the second row of the table under the equation in the top row where each fits the best.

$\sqrt{\text{25}}=5$ ; $b=\sqrt{b^2}$ ; $\sqrt{9}=3$ ; $\sqrt[6]{\text{64}}=2$ ; $\sqrt[3]{a^3}=a$ ; $\sqrt[3]{8}=2$ ;

$\sqrt[4]{\text{81}}=3$ ; $\sqrt{\text{64}}=8$ ; $\sqrt{\text{49}}=7$