+153 a.Determine the missing side length:
![[asy] size(4cm);void mra(pair u, pair v, pair w, real r){ draw(v+(u-v)/length(u-v)*r--v+(u-v)/length(u-v)*r+(w-v)/length(w-v)*r--v+(w-v)/length(w-v)*r);} pair a=(0,0); pair b=(28,0); pair c=(0,-21);draw(a--b--c--a);mra(c,a,b,2.5);label(](/api/ssl-img-proxy?src=https%3A%2F%2Flatex.artofproblemsolving.com%2F4%2Fb%2F5%2F4b5d2a1ecf68441300b385a5033077bf7da2d086.png)
b. Determine the missing side length
![[asy] size(4cm);void mra(pair u, pair v, pair w, real r){ draw(v+(u-v)/length(u-v)*r--v+(u-v)/length(u-v)*r+(w-v)/length(w-v)*r--v+(w-v)/length(w-v)*r);} pair a=(0,42/sqrt(85)); pair b=(-36/sqrt(85),0); pair c=(49/sqrt(85),0);draw(a--b--c--a);mra(c,a,b,0.75);label(](/api/ssl-img-proxy?src=https%3A%2F%2Flatex.artofproblemsolving.com%2F3%2Fb%2F4%2F3b4716eb625e93d475edde752ba453d5cba52e7b.png)
c. Determine the missing side length
![[asy] size(4cm);void mra(pair u, pair v, pair w, real r){ draw(v+(u-v)/length(u-v)*r--v+(u-v)/length(u-v)*r+(w-v)/length(w-v)*r--v+(w-v)/length(w-v)*r);} pair k=(0,0); pair l=(3,sqrt(3)); pair n=(3,0);label(rotate(30)*](/api/ssl-img-proxy?src=https%3A%2F%2Flatex.artofproblemsolving.com%2F9%2Fe%2F7%2F9e75a24f8d0e45628428c17354d15bf8451d8c04.png)
+4623 1. 28^2+21^2=c^2,\(c=35.\)
2. 6^2+7^2=c^2, c^2=85, \(c=\sqrt{85}.\)
3. \(c^2+(\sqrt{3})^2=(2\sqrt{3})^2, c^2=9, c=3.\)
+1356 Ok, all of these are relativaly simple.
The formula you will be working with for all of these is a^2+b^2=c^2
Im not going to give you the answers only explain how to do it.
#1. 28 and 21 are a and b. Insert them into the formula to get 28^2+21^2=c^2
Solve for 28^2+21^2 and get 1225
now you are at 1225=c^2
Your answer will be the square root of 1225.
#2. In this problem 6 and 7 are a and b.
Put them in the equation above to get 36+49=c^2
Your answer for this will be the square root of 85
#3. In this problem 2 times the square root of 3 is c and the square root of 3 is a.
Put them into the equation above to get the square root of 3 squared (3) + b^2=2times the square root of 3 squared (2 times the square root of 3.
This gets us to the Square root of 3+ b^2=2 times the square root of 3
Minus Square root of 3 from right side to get b^2 equals the square root of three. so b equals the square root of the square root of three
Have A nice day!
