Find The Length Of The Diagonal Of A Rectangle Formula Geometry for Beginners – How to Find the Area of a Triangle

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Geometry for Beginners – How to Find the Area of a Triangle

Welcome to another chapter of Geometry for Beginners. Our topic today is related to finding the area of ​​a triangle. For your best understanding of this topic, you need to be well versed in the previous two topics: (1) finding the area of ​​a rectangle, and (2) that area is measured with real squares, with the result labeled as square inches. or ft^2. If you don’t immediately get a sense of understanding any of this, you should first read appropriate geometry for beginners articles and get that information “implanted in your mind”. Geometry, like algebra, builds on top of previous knowledge. Without prior knowledge as a basis, new information has no meaning and cannot be learned effectively.

If you have a formula in your mind, in symbols and words, for the area of ​​a rectangle, and you understand why the area is labeled in square units, we can move on to finding the area of ​​triangles. I want to mention that while the title says we’re going to find the area of ​​a triangle – and we are – there are actually many different triangles. Rectangles differ in some ways, but opposite sides are always equal and angles are always right angles, so one side is a base. b And the other side is height h, and the formula A = bh always applies to rectangles. The shape of the triangle varies a lot, so we have to consider a few different situations. However, this is just a formula for the area of ​​a triangle in geometry, so it should be easy to remember. This is good news. What will differ is the location of the elevation. This is the part that may be the problem.

To develop the formula for the area of ​​a triangle, we first need a diagram of the rectangle. Remember to make it large enough to label the bottom with both the word “base” and the symbol b, and to label the perpendicular side with the word “Height” and the symbol h. Next to your diagram, write the formula for area in both words and symbols. “Area of ​​a rectangle is base times height” and A = bh.

Now, I want you to draw a diagonal of the rectangle on your diagram. In a rectangle, a diagonal joins opposite corners. Can you now see that the diagonals form two similar triangles? For these triangles, finding the area is fairly simple since each triangle is half of a rectangle. If the rectangle is 6 in. by 8 in. If yes, then the total area is A = bh = 6 x 8 = 48 square inches. Then the area of ​​each triangle is 24 square inches, and that leads us to the direction. Formula for area of ​​triangle: A = 1/2 bh. (Note: It can be confusing to always use A for area because it doesn’t specify what the figure is. To handle this, we sometimes use a small figure as a subscript. The area of ​​the triangle can then start with A but with a lower case. A as the subscript number. A triangle drawn on the lower right side of will be written. I don’t have that ability here, but I hope you can picture what I mean.)

In words: The area of ​​a triangle is half the base times the height.” Shortcut version: “The area of ​​a triangle is half the base times the height.” Symbol version: A = 1/2 bh .

Be careful! Be careful! Be careful! Now we get to the part where you really need to pay close attention. Remember that all rectangles have right angles, but not all triangles have right angles. When a triangle is right-angled, one leg of the right angle can be considered the base and the other leg the altitude. But what if the angle is not right?

To deal with the “no right angle” situation, I want you to look at your rectangle drawing as if you nailed some sticks together to make your rectangle. If you’ve ever done something like this you know that without some extra support pieces, the rectangle starts to bend and lose those right angles. Your rectangle will begin to look like a parallelogram with opposite sides equal and opposite angles equal. (A rectangle is actually a “special case” of a parallelogram.) Now focus on this. As we push the top corner to make our rectangle farther and farther away from the sides, the base stays the same length, but the height gets shorter. Our rectangle, 6 inches by 8 inches, becomes a parallelogram without right angles There is still an 8 inch base, but the 6 inch side height is no longer there. Area changes as elevation changes. If not 6 inches tall, what is?

I believe you notice that height is always measured as the shortest distance down to the base and this shortest distance is measured from the top vertex directly down to the base. There is actually no line visible yet, which represents height. What we do is a “straight drop” line from the top vertical to the base. This means we draw a straight line segment. its length New line is the HEIGHT of the parallelogram. The formula for area still remains the same: A = bh, but we must be very careful to choose the correct length as height. Without a right angle, the altitude is not a side of a parallelogram.

Drawing a diagonal in a parallelogram does the same thing as a rectangle–divides the figure into two equal triangles; So the area of ​​the triangle is still half of the total area. Thus, the formula for the area of ​​all triangles is the same as before: A = 1/2 bh. Again, though, we have to be careful what number we’re using for height.

On your paper, draw a non-right triangle. Label the underside with the “base” and symbol b. Find the top vertex and draw a perpendicular line straight down from that vertex to the base. Label this new line as “Height” and h. For now, you’ll only be able to find the area if you’re given height in the problem. Finding height requires skills we don’t yet have, if not given to us. For now, just remember that:

The area of ​​a triangle is the base times half the height, or A = 1/2 bh.

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