Answer:
RV ≈ 12.85, m∠V ≈ 69°, m∠R ≈ 21°
Step-by-step explanation:
As triangle RAV is a right triangle, use Pythagoras Theorem to find the measure of RV.
Pythagoras Theorem
[tex]a^2+b^2=c^2[/tex]
where:
- a and b are the legs of the right triangle.
- c is the hypotenuse (longest side) of the right triangle.
Given:
- a = AV = 4.6
- b = AR = 12
- c = RV
Substitute the given values into the formula and solve for RV:
[tex]\implies 4.6^2+12^2=RV^2[/tex]
[tex]\implies 21.16+144=RV^2[/tex]
[tex]\implies RV^2=165.16[/tex]
[tex]\implies RV=\sqrt{165.16}[/tex]
[tex]\implies RV \approx 12.85[/tex]
Use the tan trigonometric ratio to find the measure of angle V.
Tan trigonometric ratio
[tex]\sf \tan(\theta)=\dfrac{O}{A}[/tex]
where:
- [tex]\theta[/tex] is the angle
- O is the side opposite the angle
- A is the side adjacent the angle
Given:
- [tex]\theta[/tex] = V
- O = AR = 12
- A = AV = 4.6
Substitute the given values into the formula and solve for V:
[tex]\implies \sf \tan(V)=\dfrac{12}{4.6}[/tex]
[tex]\implies \sf V=\tan^{-1}\left(\dfrac{12}{4.6}\right)[/tex]
[tex]\implies \sf V\approx 69^{\circ}[/tex]
Interior angles of a triangle sum to 180°.
⇒ m∠A + m∠V + m∠R = 180°
⇒ 90° + 69° + m∠R = 180°
⇒ 159° + m∠R = 180°
⇒ m∠R = 180° - 159°
⇒ m∠R = 21°
Conclusion
RV ≈ 12.85, m∠V ≈ 69°, m∠R ≈ 21°
Learn more about Pythagoras Theorem here:
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Learn more about Trigonometric Ratios here:
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