To find the angle RPQ, we take the arctan (inverse tangent) of 1/2
RPQ=arc tan (1/2)≈26.57⁰
Given that in triangle PQR, there's a right angle at Q and side PQ is twice the length of side QR.
Let's denote:
QR as x (the shorter side)
PQ as 2x (since PQ is twice the length of QR)
In a right-angled triangle, we can use the Pythagorean theorem, which states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
So, applying the theorem:
PQ²=QR² + PR²
(2x)²=x² + PR²
4x² = x² + PR²
3x² = PR²
Now, to find the ratios of the sides in a right-angled triangle, we use trigonometric ratios. In this case, we're looking for the tangent of angle RPQ.
tan(RPQ) = opposite side / adjacent side = QR/PQ
x/2x=1/2
So, tan(RPQ)=1/2
To find the angle RPQ, we take the arctan (inverse tangent) of 1/2
RPQ=arc tan (1/2)≈26.57⁰