1. Module 6

1.24. Page 4

Mathematics 10-3 Module 6 Lesson 5

Module 6: Triangles and Other Polygons

 

Bringing Ideas Together

 

In the Explore section you proved the Pythagorean Theorem. The statement of the theorem applied to a right triangle is labelled below. The length of the hypotenuse is c. The lengths of the legs are a and b.

 

This illustration shows a right triangle with legs a and b and hypotenuse c. There are blue squares on the legs with areas a squared and b squared. There is a yellow square on side c with area c squared.

 

Remember, the Pythagorean Theorem only applies to right triangles, as you discovered in the Get Started section. It states that for any right triangle, the square on the hypotenuse equals the sum of the squares on the two legs.

 

The letters used in the statement depend on the labels on the sides of the triangle.

 

Example 2

 

Write the statement of the Pythagorean Theorem for each right triangle.

 

  1. This illustration shows right triangle PQR. The side opposite angle R is labelled r, the side opposite angle P is labelled p. The side opposite angle Q is labelled q.

  2. This illustration shows right triangle ABC. The side opposite angle A is labelled s. The side opposite angle B is labelled b. The side opposite angle C is labelled c.

Solution

  1. Notice that, here, the lengths of the sides of the triangle are named using lowercase (small) letters. The letter used depends on the opposite angle.

    The length of side QR is p, because this side lies opposite ∠P.
    The length of side PR is q, because this side lies opposite ∠Q.
    The length of side PQ is r, because this side lies opposite ∠R.

    Because q is the length of the hypotenuse, q2 = p2 + r2, or p2 + r2 = q2.

  2. Once again, the lengths of the sides of the triangle are named using lowercase (small) letters. Again, the letter used depends on the opposite angle.

    The length of side BC is a, because this side lies opposite ∠A.
    The length of side AC is b, because this side lies opposite ∠B.
    The length of side AB is c, because this side lies opposite ∠C.

    Because b is the length of the hypotenuse, b2 = a2 + c2, or a2 + c2 = b2.
The Carpenters’ Corner

 

In Get Started, you stretched a paper loop to form a 3-4-5 right triangle. This Pythagorean triple is used by carpenters to check that corners are square. Carpenters will mark points 3 ft and 4 ft from the corner.

 

This illustration shows the corner of a room with a mark on one wall that is 3 feet from the corner and a mark on the other wall that is 4 feet from the corner. The line joining these points is labelled “Is this 5 feet?”

 

If the distance between the marks is 5 ft, the corner is square.

 

Because triangles with proportional sides are similar, the following side-length ratios will all be Pythagorean triples!

 

 

 

 

 

 

3:4:5

6:8:10

9:12:15

12:16:20

 

Do you see the pattern here? How would you find another Pythagorean triple with this same ratio?

 

m10_3_selfcheck.jpg Self-Check

 

SC 3.

  1. Show that 5, 12, 13 is a Pythagorean triple.
  2. Sketch a triangle with these sides.
  3. From 5, 12, 13, write three more Pythagorean triples.

SC 4.

  1. Is 4, 7, 9 a Pythagorean triple?
  2. Sketch a triangle with these sides.
  3. Is the triangle a right triangle? Why or why not?

Compare your answers.

 

Example 3

 

A ladder is leaning against a vertical wall. The foot of the ladder is 1 m from the wall, and the ladder reaches 3 m up the wall. How long is the ladder? Round to 1 decimal place.

 

This illustration shows a ladder leaning against a wall. The ladder reaches 3 metres up the wall and its feet are 1 metre from the wall. The ladder, the floor, and the wall form a right triangle.


Solution

 

Let the length of the ladder be x.

 

 

 

 

The ladder is about 3.2 m in length.

 

m10_3_selfcheck.jpg Self-Check

 

SC 5. Kale walked 300 m north, turned, and then walked 200 m east. How far is Kale from his starting point? Round your answer to the nearest metre.

 

Compare your answers.

 

m10_3_mastcon.jpg Mastering Concepts

 

Try this question. When you are finished, check your answer.

 

The numbers 3, 4, and 5 form a Pythagorean triple. Prove that the triplet 3n, 4n, 5n, where n is any positive whole number (n = 1, 2, 3, or 4, and so on), is also a Pythagorean triple.

 

Compare your answer.