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When one side of triangle is makes an angle of 90 degrees, then given triangle is called as a Right Triangle means each right triangle has 90 degree angle. As we all know that geometric Mean of 2 Numbers are √ (a * b), where ‘a’ and ‘b’ are two numbers. Now we will discuss how to find out geometric mean in right Triangles.
For calculating Altitude of right triangle, we use geometric mean between 2 hypotenuses, so geometric mean in right triangle is,
Altitude of right triangle = √ (length of hypotenuse1 * length of hypotenuse2),
Suppose we have two hypotenuse length of right triangle is 4 inch and 6 inch, then altitude of right triangle is –
Altitude of right triangle = √ (length of hypotenuse1 * length of hypotenuse2),
= √ (6 * 4),
= √ 24,
= 4.94.
So, altitude of right is 4.94 inch, where other 2 hypotenuse length is 6 inch and 4 inch.
Altitude of right triangle divides triangle into two parts, where altitude is part of both sides like we have a right triangle ABC, where ‘BD’ is an altitude of given right triangle, then by altitude geometric mean right triangles theorem–
AD / BD = BD / CD,
= > (BD)^{2} = AD * CD,
So, altitude BD is behaving like geometric mean of given hypotenuse length of right triangle, which is AD and CD means AD and CD both are hypotenuse length of right triangle.
Suppose we have a right triangle PQR, where PS is an altitude and hypotenuse length of RS = 8 inch and QS = 6 inch, then,
Here we apply geometric mean theorem for evaluation of altitude PS–
PS = √ (length of RS * length of QS),
= √ (8 * 6),
= √ 48,
= 6.91.
So, altitude of right triangle PQR is 6.91 inch.

In mathematics, Pythagorean Theorem is used in many proofs which help in solving the various lengths of various geometrical shapes. It is used to calculate the length of different sides of right angled triangle or that shapes, which are based on these types of shapes. It can also be find by the geometric proof as well as algebraic proof. It not only defined for Euclidean figures but also for n – dimensional shapes that are Solid in form.
Now we will proceed to Altitude to the hypotenuse.
In the given figure 1, the Right Triangle PQR has altitude QS that is drawn to the hypotenuse PR.
By using the AA Similarity Postulate this theorem cannot be shown easily.
Now we will see the Theorem of altitude to the hypotenuse. In the given figure, the altitude plot to the hypotenuse of a right triangle which creates two similar right Triangles, in the given triangles each are parallel to the original right triangle and analogous to one another.
Figure 2 defines the three right triangles created in Figure 1. All these triangles are drawn in such a way that equivalent parts of triangle are easily known.
In the given figure ‘PQ’ and ‘QR’ are legs of the original right triangle; PR is the hypotenuse in the original right triangle; QS is the altitude drawn to the hypotenuse; PS is the segment on the hypotenuse touching leg PQ and SR is the segment on the hypotenuse touching leg QR.
As we know that these all triangles are related to each other, and the ratios of all pairs of corresponding sides are same.