Answer:
[tex]h=6\\k=2.5\\n=3\sqrt{10}\\m=\sqrt{21}\\p=\sqrt{17}[/tex]
Step-by-step explanation:
All of the triangles in this problem are right triangles. This means that all of them contain a right angle, this is indicated by the box around one of the angles. The Pythagorean theorem is a property of the sides of a right triangle, this property, in the form of an equation, is as follows,
[tex]a^2+b^2=c^2[/tex]
Where (a) and (b) are the legs, that is, the sides that are adjacent to the right angle of a right triangle. The parameter (c) represents the hypotenuse, the side opposite the right angle of a right triangle. For each triangle, substitute the given side lengths, simplify, and solve for the unknown side.
[tex](h)^2+(8)^2=(10)^2\\h^2+64 = 100\\h^2=36\\h=6[/tex]
[tex](k)^2+(6)^2=(6.5)^2\\k^2+36=42.25\\k^2=6.25\\k=2.5[/tex]
[tex](n)^2+(\sqrt{10})^2=(10)^2\\n^2+10=100\\n^2=90\\n=\sqrt{90}\\n=3\sqrt{10}[/tex]
[tex](m)^2+(2)^2=(5)^2\\m^2+4=25\\m^2=21\\m=\sqrt{21}[/tex]
[tex](p)^2+(\sqrt{68})^2=(\sqrt{85})^2\\p^2+68=85\\p^2=17\\p=\sqrt{17}[/tex]
