**Middle School Number Sense Lesson 53: Triangles**

We have already covered right triangles in

**TriangleRight**; today we will discuss properties of other types of triangles. This concept appeared

**16 times**last year, with a median placement at

**question # 53**.

**Number Dojo Level: 148**

Let's start with a few definitions:

**Equilateral**triangle: has 3 equal sides & 3 equal angles**Isosceles**triangle: has 2 equal sides & 2 equal angles**Scalene**triangle: has 0 equal sides & 0 equal angles**Vertex**angle: the top (non-equal) angle of an isosceles triangle which is resting on its non-equal side**Base**angle: the bottom (equal) angle of an isosceles triangle which is resting on its non-equal side

And a few reminders:

- The
**area**of a triangle is**bh/2**(1/2 half the product of its length & height) - The
**perimeter**of a triangle is the sum of its sides, or**3s**for an equilateral triangle - The sum of the
**interior angles**of any triangle is 180°.

And a few formulas specific to equilateral triangles:

- A = (√3/4)ss = (√3/3)hh
- h = (√3/2)s
- P = (2√3)h

**Example 1: Find the perimeter of an equilateral triangle with side of 433.**

- P = 3s (for equilateral triangles) = 3(433)
- 3 x 433 =
**1299**.

**Example 2: The vertex angle of an isosceles triangle with base angle 48° is ___°**

- The sum of the interior angles of any triangle is 180°. The two base angles (48°) are equal.
- 180 - (2 x 48) = 180 - 96 =
**84**.

**Example 3: If the area of an equilateral triangle with side 8 is k√3, then k = ___**

- A = (√3/4)ss
- √3/4 x 8 x 8 = √3/4 x 64 =
**16√3**. k =**16**.

**Example 4: The area of an equilateral triangle is 12√3 sq inches. Its height is ___ inches.**

- A = (√3/3)hh
- 12√3 = (√3/3)hh. Divide both sides by √3 to get
**12 = hh/3**. - Multiply both sides by 3 to get
**36 = hh**. - Take the square root of both sides to get
**h = 6**.

**Example 5: A triangle has integral sides of 3, x, and 9. The largest value for x is ___**

- To me, this is more of a logic problem than a math problem. The idea is that
**no side length can be equal to or greater than the sum of the other two sides**. - If x can't be equal to or greater than (3 + 9), then its largest integral value is (3 + 9 - 1), which is
**11**.

**Example 6: When the height of a triangle with base 11 is increased from 14 to 32, the corresponding increase in area is ___**

- Let's try calculating the area of each triangle and finding the difference.
- A1 = (11)(14)/2 = 11(7) =
**77**. - A2 = (11)(32)/2 = 11(16) =
**176**. - 176 - 77 =
**99**. - Now let's try (11)(32 - 14)/2 = (11)(18)/2 = 11(9) =
**99**. This may be a time-saver!

**Here's a free worksheet to help you practice Triangles:**

triangles.pdf |

**Up Next for Middle School: PolygonDiag**