With all this talk about triangles and their side lengths and angles, we shouldn't forget to discuss how to find the AREA of a triangle.
You are probably familiar with one formula for finding the area of a triangle:
Area = 1/2 (base)(height)Compare this to finding the area of a rectangle:
The area of the rectangle is equal to the product of (base) x (height)..... (or length x width). However, by drawing a diagonal within the rectangle which joins two opposite corners, you can see that each newly-formed triangle is equal to half of the area of the original rectangle. Therefore, the area of a triangle is one-half the area of the rectangle, as shown by this triangle area formula. Even if you are looking at a triangle that doesn't immediately look like it is half of a rectangle, this formula still applies.
To prove it, you can draw a line in to represent the height, as I have shown here, thus creating two smaller triangles, and you can rearrange them to see that they indeed are equal to the area of half a rectangle:
That is one way to find the area of a triangle. However, if instead of base and height measurements, you are given lengths of sides or angles, this method won't work for you. In this case, you need to use a trig equation to solve for the area of a triangle.
Let's start with the first equation we had above, and modify it. By the standard trig identities, we can show that:
height = (a)(SinC)So substituting that into our formula:
Area = 1/2(base)(height)
Area = 1/2(b)(a)(SinC)
And this is the trig formula for solving the area of a triangle!
Area = (1/2)abSinCYou can use this to find the area of a triangle where you know any two sides and the angle between them! It's that easy!
3 comments:
I have been finding it difficut to explain the concept of "least common multiple and the greatest common divisor" to my 11 years old kid. I would preferably want to explain it from practical application perspective. Suggestions would be very useful. Also your thoughts on the relationship between fractions and ratios.
"Least common multiple" (LCM) is the lowest number that is a multiple of two other numbers. For example, for the numbers 3 and 4, the LCM is 12:
3x1=3
3x2=6
3x3=9
*3x4=12*
4x1=4
4x2=8
*4x3=12*
Similarly, "greatest common divisor" is the largest number that can divide *evenly* into two other numbers. For example, for 12 and 18, the greatest common divisor is 6.
12/12=1
*12/6=2*
12/2=6
12/1=12
18/18=1
18/9=2
*18/6=3*
18/3=6
18/2=9
18/1=18
In terms of a practical application perspective, these are important concepts in computer science and programming, although I don't know how enlightening that would be for an 11 year old. :)
As for fractions and ratios, they are similar, but there is an important distinction to make. Fractions are essentially a piece of a whole, eg. 3/4 = 3 pieces of pie out of total 4 pieces. However, for a ratio, you are not dealing with the whole, but rather a comparison of two sets of pieces, eg. 3:1 = 3 pieces of apple pie to 1 piece of cherry pie. The total number of pieces can be found by adding the 2 numbers in the ratio, eg. 4 pieces.
Hope that helps!
A little late on this I know, but I would think in trying to teach LCM, a student needs to be used to the meaning of the term "multiple" or "multiples". Multiples are used when we count by any number. Hence, multiples of 4 are, 4,8,12,16,20 etc. Students must quickly think: "oh, multiples, that means I'm counting by that number." Play this game with your kid: OK, Tell me the first 7 multiples of 6. or OK, what is the 6th multiple of 4?
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