Question

Precal : Note: Triangle may not be drawn to scale. Suppose a = 7 and c = 12 .

Answer 1

Step-by-step 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:

`h^2=a^2+b^2\Rightarrow b^2=h^2-a^2\Rightarrow b=√(h^2-a^2).`

Where h is the hypotenuse, a and b are the cathetus.

Replacing the values from the statement, we get:

`b=√(12^2-7^2)=√(144-49)=√(95)\cong9.7.`

(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:

`\sin(\theta)=(oc)/(h).`

Replacing the data from above, we have:

`\sin(A)=(7)/(12)\Rightarrow A=\sin^(-1)((7)/(12))\cong35.7\degree.`

(3) From geometry, we know that the inner angles of a triangle sum 180°, so we have:

`A+B+90\degree=180\degree\Rightarrow B=180\degree-90\degree-A=90\degree-A.`

Replacing the value obtained for A, we get:

`B\cong90\degree-35.7\degree=54.3\degree.`Answer

• b = ,9.7

,

• A = ,35.7, degrees

,

• B = ,54.3, degrees