Explanation
We have a triangle with:
• cathetus a = 7,
,
• cathetus b = ?,
,
• hypotenuse h = c = 12.
(1) Pitagora's Theorem states that for a right triangle:
[tex]h^2=a^2+b^2\Rightarrow b^2=h^2-a^2\Rightarrow b=\sqrt{h^2-a^2}.[/tex]
Where h is the hypotenuse, a and b are the cathetus.
Replacing the values from the statement, we get:
[tex]b=\sqrt{12^2-7^2}=\sqrt{144-49}=\sqrt{95}\cong9.7.[/tex]
(2) We consider the right triangle with:
• angle θ = A,
,
• opposite cathetus oc = a = 7,
,
• hypotenuse h = c = 12.
From trigonometry, we know the relation:
[tex]\sin(\theta)=\frac{oc}{h}.[/tex]
Replacing the data from above, we have:
[tex]\sin(A)=\frac{7}{12}\Rightarrow A=\sin^{-1}(\frac{7}{12})\cong35.7\degree.[/tex]
(3) From geometry, we know that the inner angles of a triangle sum 180°, so we have:
[tex]A+B+90\degree=180\degree\Rightarrow B=180\degree-90\degree-A=90\degree-A.[/tex]
Replacing the value obtained for A, we get:
[tex]B\cong90\degree-35.7\degree=54.3\degree.[/tex]Answer
• b = ,9.7
,
• A = ,35.7, degrees
,
• B = ,54.3, degrees
