Question 825187: if the sides fo triangle are 5 cm 12 cm and 13 cm respectively , what is the length of perpendicular form the opposite vertex to the longest side?
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! A key to this problem is to recognize that the initial triangle is a right triangle. When a triangle is a right triangle the three sides fit the Pythagorean equation,  . It also works in reverse: If the three sides of a triangle fit the equation, then the triangle must be a right triangle. So, since  , the given triangle is a right triangle.
Next, it might help to draw a diagram. Draw a right triangle and label the legs as 5 and 12 and the hypotenuse as 13. Then draw a "perpendicular form the opposite vertex to the longest side". Since this is a right triangle, the longest side is better known as the hypotenuse and the vertex opposite the hypotenuse is the vertex where the right angle is.
Looking at this diagram you might recognize it. It is a larger right triangle, the one with sides of 5, 12 and 13, with two smaller right triangles inside of it. This type of diagram is often used when working with similar triangles because... all three triangles in the diagram are similar to each other! (I'm not going to go into the details of proving this. If you're curious then try to find two pairs of angles, one angle from each triangle in each pair, that are congruent. This will prove similarity by the AA postulate of similarity.)
When polygons, like triangles, are similar then every ratio of corresponding sides are equal to each other. Let's look at the ratio of the short leg to the hypotenuse:
In the large triangle this ratio would be 5/13. And if we call the length of the perpendicular we drew "x", then the ratio of the short side to the hypotenuse of the triangle whose hypotenuse is 12 is: x/12. Because of the similarity of the two triangles these ratios must be equal. So:
We can solve this for x. Cross-multiplying we get:
5*12 = 13*x
which simplifies to:
60 = 13x
Dividing by 13:
Since "x" is what we were asked to find, this is our answer.
P.S. It might be good to try to remember a diagram of three right triangles (like we had in this problem) has three similar right triangles in it.
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