Right Triangles TC2MA234

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)

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?

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

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 = 32 + 42.

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.

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

Find the length of the hypotenuse if a = 12 and b = 16. 122 + 162 = c2 144 + 256 = c2 400 = c2 Take the square root of both sides. 20 = c

Find the length of the hypotenuse if a = 5 and b = 7. 52 + 72 = c2 25 + 49 = c2 74 = c2 Take the square root of both sides. 8.60 = c

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

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

What’s the area? 8 cm 32 cm2 8 cm

What’s the area? 8 cm 12 cm2 3 cm

Proof of Pythagoras theorem:

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!

Special Right Triangles

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.

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

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

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

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

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°

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.

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