How do you derive the formula for a triangle?

Answer 1

Probably, the question is about a formula for an area of a triangle.
If not, ask more specifically.
Below is the proof of a formula #S=1/2*a*h#,
where #a# is a base and #h# is an altitude.

Consider triangle #Delta ABC# below

An altitude #h# separates our triangle into two right triangles.
As seen from the picture, the area of each one of these right triangles is doubled, so the area of a rectangle with the same base #a# and altitude #h# is twice the area of an initial triangle.

Since the area of a rectangle is #a*h#, the area of the initial triangle is
#S = 1/2*a*h#

The drawing will be different in case of a triangle with an obtuse angle at the base. In this case an analogous transformation into a rectangle will represent our triangle not as a sum of two right triangles, but as their difference with the same final formula.

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Answer 2

The formula for the area of a triangle can be derived using basic principles of geometry. One common method is to use the formula for the area of a parallelogram and then halve it, since a triangle is half of a parallelogram.

  1. Start with a parallelogram with base ( b ) and height ( h ). The area of a parallelogram is given by ( A = bh ).

  2. Draw a diagonal line from one vertex of the parallelogram to the opposite vertex, creating two congruent triangles.

  3. The area of one of these triangles is half the area of the parallelogram, so the area of one triangle is ( \frac{1}{2}bh ).

  4. Since any triangle can be divided into two congruent triangles, the area of any triangle is also given by ( \frac{1}{2}bh ), where ( b ) is the base and ( h ) is the corresponding height (perpendicular distance from the base to the opposite vertex).

Therefore, the formula for the area of a triangle is ( \frac{1}{2} \times \text{base} \times \text{height} ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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