Proof 3.7c

Author:: Kayla Moore

Proof: Consider the right

with a right angle at

(Definition 21). Construct similar triangles on the sides of

. Notice, by construction,

and

. We know that the area of a triangle can be found using the formula,

. To find the area of

, we would use the length of

times length of

times

. To find the area of

, we would use the length of

times

. To find the area of

, we would use the length of

times length of

times

. Since

is congruent to

, we know that the areas must also be equal by Common Notion 1. Since

is composed of two smaller triangles, we can their two ares together to find the whole area of the triangle. To do this, we would add the areas of

and

. We know by construction,

, so this value can be substituted. In this case, the area is

because of similar triangles as discussed with Ceva's Theorem and the addition properties previously discussed. If we complete the algebra, as follows:

(if we multiply everything by

) Therefore, we can conclude, by using similar triangles, that

when a and b are legs of a right triangle and c is the hypotenuse.

Proof 3.7c

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