Question

Find the value of each variable for the right triangles. Make sure all answers are in reduced radical from. Shoe your work.

Answer 1

Denote each of the triangles ABC, where B is the right angle.

a.

$t^{2}+ 12^{2}= 13^{2}$

instead of taking the square of 12 and 13, , take 12 squared to the right side and apply the difference of squares formula: $a^{2}- b^{2}=(a-b)(a+b)$

$t^{2}= 13^{2}- 12^{2}=(13-12)(13+12)=1*25$

so t is square root of 25, which is 5

b. 

$a^{2}+9 ^{2} = 12^{2}$

$a^{2}= 12^{2} -9 ^{2}=(12-9)(12+9)=3*21=3*3*7$

$a= \sqrt{3*3*7}=3 \sqrt{7}$


b.

$6^{2} + 9^{2} = x^{2}$

$36+81= x^{2}$

here we can factorize 9, to work with small numbers:

$9(4+9)= x^{2}$

$9*13= x^{2}$

$x^{2} =3*3*13$

$x= \sqrt{3*3*13}=3 \sqrt{13}$