50 points pl help 12. A triangle has side lengths of 6, 8, and 10. Is it a right triangle? Explain.

Step-by-step explanation:

We are given three sides of the triangle and we need to say that whether the triangle is right Angled triangle or not . The given side lenghts are 6 cm , 8 cm and 10 cm . ( Units not given in Question ) .

So , a triangle with its given sides will be a right angled triangle if it has a right angle . And the sum of squares of two smallest sides must be equal to the square of the longest side . ( According to Pythagoras Theorem ) .

Here two smallest sides are 8cm and 6cm .

And the largest side is 10 cm .

And here the sum of squares of 8cm and 6cm should be equal to the square of 10cm in order to Prove it a right angled triangle .

Since LHS = RHS hence the triangle is a right angled triangle .

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4