1. Right Triangles
TC2MA234

2. What is a right triangle?
(altitude) side
hypotenuse
right angle
It is a triangle which has an angle that is 90 degrees.
The two sides that make up the right angle are called legs (base & altitude).
The side opposite to the right angle is the hypotenuse.
side (base)

3. Which gold plate would you choose?
Suppose I were to beat gold into place as shown below, what option would you choose?
Plate c2
Plate a2 and plate b2?

4. Pythagorean Theorem c2 = a2 + b2
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
c2 = a2 + b2

5. Using the Pythagorean Theorem
A Pythagorean triple is a set of three positive integers a, b, and c that satisfy the equation c2 = a2 + b2 For example, the integers 3, 4 and 5 form a Pythagorean Triple because 52 =

6. Ex. 1: Finding the length of the hypotenuse.
Find the length of the hypotenuse of the right triangle. Tell whether the sides lengths form a Pythagorean Triple.

7. Ex. 2: Finding the Length of a Leg
Find the length of the leg of the right triangle.

8. Find the length of the hypotenuse if a = 12 and b = 16.
= c2
= c2
400 = c2
Take the square root of both sides.
20 = c

9. Find the length of the hypotenuse if a = 5 and b = 7.
= c2
= c2
74 = c2
Take the square root of both sides.
8.60 = c

10. A = ½ bh Area of Triangles h b
The area of a triangle is half of the area of a parallelogram.
A = ½ bh

11. What’s the area?
4 cm
8 cm
16 cm2
½ of 8 x 4 =

12. What’s the area?
8 cm
32 cm2
8 cm

13. What’s the area?
8 cm
12 cm2
3 cm

14. Proof of Pythagoras theorem:

15. Proof: Area of the square = (a + b)2 = a2 + b2 + 2ab
The area of the square is also the area of the pink square as well as the 4 blue triangles.
Area of the pink square = c2
Area of the blue triangles = ½ ab Area of 4 triangles = 4(½ ab) = 2ab
Therefore total area = c2 + 2ab
(iii) = (1) Therefore , a2 + b2 + 2ab = c2 + 2ab or a2 + b2 = c2 Hence the proof!

16. Special Right Triangles

17. Side lengths of Special Right Triangles
Right triangles whose angle measures are 45°-45°-90° or 30°-60°-90° are called special right triangles. The theorems that describe these relationships of side lengths of each of these special right triangles follow.

18. 45°-45°-90° Triangle Theorem
In a 45°-45°-90° triangle, the hypotenuse is √2 times as long as each leg.
45°
√2x
45°
Hypotenuse = √2 ∙ leg

19. 30°-60°-90° Triangle Theorem
In a 30°-60°-90° triangle, the hypotenuse is twice as long as the shorter leg, and the longer leg is √3 times as long as the shorter leg.
60°
30°
√3x
Hypotenuse = 2 ∙ shorter leg
Longer leg = √3 ∙ shorter leg

20. Ex. 1: Finding the hypotenuse in a 45°-45°-90° Triangle
Hypotenuse = √2 ∙ leg
x = √2 ∙ 3
x = 3√2 3 3
45° x
45°-45°-90° Triangle Theorem
Substitute values
Simplify

21. Ex. 2: Finding a leg in a 45°-45°-90° Triangle
Find the value of x.
Because the triangle is an isosceles right triangle, its base angles are congruent. The triangle is a 45°-45°- 90° right triangle, so the length of the hypotenuse is √2 times the length x of a leg. 5 x x

22. Ex. 3: Finding side lengths in a 30°-60°-90° Triangle
Find the values of s and t.
Because the triangle is a 30°-60°-90° triangle, the longer leg is √3 times the length s of the shorter leg.
60°
30°

23. Converse of the Pythagorean theorem:
If Δ ABC is a triangle with sides of lengths a, b, and c such that a2 + b2 = c2; then ΔABC is a right angled triangle with the right angle opposite to the side c.

24. Exercise
Determine if the following lengths can be the sides of a right triangle:
51, 68, 85
2, 3, √13
3, 4, 7