?if pqr is a triangle, right angled at p. if pq is 10cm and Pr is 24 c...
Given information:
- Triangle PQR is a right-angled triangle, with angle P being the right angle.
- Side PQ measures 10 cm.
- Side PR measures 24 cm.
To find:
The length of side QR.
Solution:
1. Using the Pythagorean theorem:
The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, we can use the Pythagorean theorem to find the length of side QR.
Let's denote the length of side QR as x.
According to the Pythagorean theorem, we have:
PR² = PQ² + QR²
Substituting the given values, we get:
24² = 10² + x²
Simplifying the equation:
576 = 100 + x²
Rearranging the equation:
x² = 576 - 100
Calculating:
x² = 476
Taking the square root of both sides:
x = √476
Simplifying:
x ≈ 21.86 cm
Therefore, the length of side QR is approximately 21.86 cm.
2. Applying the properties of right-angled triangles:
Alternatively, we can use the properties of right-angled triangles to find the length of side QR.
In a right-angled triangle, the lengths of the two legs (sides adjacent to the right angle) are related to each other and the hypotenuse by the following trigonometric ratios:
- Sine (sin): sin(angle) = opposite/hypotenuse
- Cosine (cos): cos(angle) = adjacent/hypotenuse
- Tangent (tan): tan(angle) = opposite/adjacent
Since angle P is the right angle, we can use the sine ratio to find the length of side QR.
sin(angle) = opposite/hypotenuse
sin(P) = QR/PR
Substituting the given values, we get:
sin(90°) = QR/24
Since sin(90°) = 1, we have:
1 = QR/24
Simplifying the equation:
QR = 24
Therefore, the length of side QR is 24 cm.
However, this solution assumes that angle P is exactly 90 degrees. If angle P is not exactly 90 degrees, the length of side QR will be different.