Question

Please help me with these geometry questions

Answer 1

a. 30 b. indeterminate, unless you're given the measure of either of the other sides.

Explanation:

a. lazy solution: it's a 3-4-5 right triangle scaled up (18 is 3*6, 24 is 4*6, last side will be 5*6= 30. Less than lazy solution: it's a right triangle, pythagorean theorem applies:

`x^2= 18^2+24^2 = 324+576 = 900\\x= 30`

b.problem is indeterminate as long as you don't know the measure of any of the other sides. the 30-60 triangle is half of an equilateral triangle and x is
`\sqrt3` times the short side, or
`\frac{\sqrt3} 2` times the long side. It's easily shown by mirroring the triangle along the vertical side, and if the short leg measures k, the hypotenuse measures 2k, and the vertical side measures
`x^2+k^2=(2k)^2\\x^2+k^2 = 4k^2\\x^2 = 3k^2 \implies x=\sqrt3\ k`