Answer:
a) p = 21.4 cm (nearest tenth)
B) P = 45.6° (nearest tenth)
Step-by-step explanation:
As triangle PQR is a right triangle, we can use Pythagoras Theorem to find the length of p.
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
From inspection of the triangle:
- a = PQ = 21
- b = QR = p
- c = PR = 30
Substitute the given values into the formula and solve for p:
[tex]\implies 21^2+p^2=30^2[/tex]
[tex]\implies 441+p^2=900[/tex]
[tex]\implies 441+p^2-441=900-441[/tex]
[tex]\implies p^2=459[/tex]
[tex]\implies \sqrt{p^2}=\sqrt{459}[/tex]
[tex]\implies p=\pm 3\sqrt{51}[/tex]
As p is the length, p can be positive only.
Therefore, p = 21.4 cm (nearest tenth)
To find the angle P, use the cosine trigonometric ratio.
Cosine trigonometric ratio
[tex]\sf \cos(\theta)=\dfrac{A}{H}[/tex]
where:
- [tex]\theta[/tex] is the angle
- A is the side adjacent the angle
- H is the hypotenuse (the side opposite the right angle)
From inspection of the triangle:
- [tex]\theta[/tex] = P
- A = PQ = 21
- H = PR = 30
Substitute the given values into the formula and solve for P:
[tex]\implies \sf \cos P=\dfrac{21}{30}[/tex]
[tex]\implies \sf P=\cos^{-1}\left(\dfrac{21}{30}\right)[/tex]
[tex]\implies \sf P=45.572996^{\circ}[/tex]
Therefore, the measure of angle P is 45.6° (nearest tenth).
Learn more about Pythagoras Theorem here:
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