prove that if the lengths of two sides of a and b, triangle are a and b respectively, then the lengths of the corresponding altitudes to those sides are in the ratio b/a.
To prove that the lengths of the corresponding altitudes to the sides of a triangle are in the ratio of the lengths of those sides, we can use the formula for the area of a triangle.
The formula for the area of a triangle is given by:
Area = (base * height) / 2
Let's consider a triangle with sides a, b, and c, where a and b are the lengths of two sides, and c is the length of the remaining side. Let h1 and h2 be the lengths of the corresponding altitudes to sides a and b, respectively.
Using the formula for the area of a triangle, we can write the following equations:
Area = (base * height) / 2
Area = (a * h1) / 2 -- (1)
Area = (b * h2) / 2 -- (2)
Since the area of a triangle remains constant, we can set equation (1) equal to equation (2):
(a * h1) / 2 = (b * h2) / 2
Now, cancel out the fraction:
a * h1 = b * h2
To prove that the lengths of the altitudes are in the ratio of b/a, we need to show that h2 / h1 = b / a.
Divide both sides of the equation by (a * h1):
h2 / h1 = b / a
Hence, we have derived that the ratio of the lengths of the corresponding altitudes to the sides of a triangle is equal to the ratio of the lengths of those sides, i.e., h2 / h1 = b / a.
Therefore, the statement is proven.