Triangles

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.

1. P = 3s (for equilateral triangles) = 3(433)
2. 3 x 433 = 1299.

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

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

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

1. A = (√3/4)ss
   
2. √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.

1. A = (√3/3)hh
2. 12√3 = (√3/3)hh.  Divide both sides by √3 to get 12 = hh/3.
3. Multiply both sides by 3 to get 36 = hh.
4. 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 ___

1. 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.
2. 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 ___

1. Let's try calculating the area of each triangle and finding the difference.
2. A1 = (11)(14)/2 = 11(7) = 77.
3. A2 = (11)(32)/2 = 11(16) = 176.
4. 176 - 77 = 99.
5. 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
File Size:	376 kb
File Type:	pdf

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Up Next for Middle School: PolygonDiag