## Triangles Lesson Part 2 : What Can I Do With Triangles?

As we mentioned, triangles are useful in engineering and design. Trigonometry, which we’ll get to in a later unit, is the study of right triangles. Trigonometry allows us to measure the distance to far away galaxies, predict the arc of a baseball, and understand waves of all kinds. The more abstract idea of a triangle is used in popular culture and language (think of love triangles and plot diagrams). Triangles are, actually, everywhere!

## Triangles Lesson Part 3 : Guided Practice

The diagram below illustrates the Pythagorean theorem, which states that the area of square c is equal to the areas of squares a + b.

1. What type of triangle is formed by the intersection of squares a, b, and c?

The triangle shown is a right triangle because one of its angle measurements is equal to 90 degrees. A, b, and c are different lengths, so the triangle is scalene. It is therefore a scalene right triangle.

2. If side a = 5 inches, b = 12 inches, and c = 13 inches, what are the perimeter and area of the triangle formed by the intersection of squares a, b, and c?

To find the perimeter of a triangle, add all of the side lengths together. The perimeter of the triangle formed by the intersection of squares a, b, and c is equal to 5 + 12 + 13 = 30 inches. To find the area, we will solve $$\frac{1}{2}*b*h$$, where b = the base and h = the height (the base and the height are always perpendicular). The base and height of the triangle are 5 and 12, therefore the area is $$\frac{1}{2}*5*12$$ = 30 inches.

3. If each of the sides of square c is equal to 13 inches, what is the length of diagonal xy?

Since the sides of a square are equal lengths, and all of the angles in a square measure 90 degrees, the diagonal of square c is the hypotenuse of a right triangle with half the area of square c. The diagonal of the square bisects (cuts in half) two of the 90 degree angles, and both of the bases of the triangle are 13 inches because they are sides of the square. This means that the triangle formed by the diagonal xy is an isosceles 45-45-90 triangle. The side ratios of a 45-45-90 triangle are $$1 : 1 : \sqrt{2}$$. If each of the bases of the triangle are equal to 13, then the diagonal is equal to $$13\sqrt{2}$$.

4. What relationship does a triangle with side measures of 10, 24, and 26 have to the triangle formed by the intersection of squares a, b, and c?

The square formed by the intersection of squares a, b, and c is a 5, 12, 13 right triangle. A 10, 24, 26 triangle is proportional to a 5, 12, 13 triangle (all of the side lengths are multiplied by 2). Therefore, the triangles are similar.

Think you’ve got the hang of it? Try out our 5-question quiz below!