More Triangle Problems

Category: Plane Geometry, Algebra

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A right triangle has an area of 54 in². The product of the lengths of the three sides is 1620 in³. What are the lengths of its sides?

Solution:

To illustrate the problem, it is better to draw the figure as follows

Photo by Math Principles in Everyday Life

We know that the area of a right triangle is

If the product of the sides of a triangle is 1620 in³, then we can solve for the length of hypotenuse as follows

Use Pythagorean Theorem in order to solve for the other sides of a right triangle as follows

To solve for a and b, we need to use another equation which is 

or

Substitute the value of b to the above equation, we have

Multiply both sides of the equation by a², we have

Use Quadratic Formula in order to solve for a² as follows

If we choose the positive sign,

The other side of a right triangle is

If we choose the negative sign,

The other side of a right triangle is

Therefore, the three sides of a right triangle are 9, 12, and 15.