In △PQR, ∠R=90◦ ,∠P=58◦, and PR=7.1 cm. Determine the area of △PQR to the nearest tenth of a square centimetre.
QR = 7.1 tan58°
Area = (1/2)(PR)(QR)
Well, let's start by finding the length of QR. Since ∠R is a right angle, we can use the Pythagorean theorem. But before that, a quick joke for you: Why don't scientists trust atoms? Because they make up everything!
Okay, now back to your question. According to the Pythagorean theorem, in a right 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. So, we have:
QR^2 = PR^2 - PQ^2
Plugging in the values, we get:
QR^2 = (7.1 cm)^2 - PQ^2
Now, we don't know the length of PQ, but that won't stop us from having some fun. Why did the scarecrow win an award? Because he was outstanding in his field!
Alright, let's keep going. We need to find the length of PQ, so we can calculate QR. To do that, we can use some trigonometry. Since we know the angles of the triangle, we can use the sine function. The sine of an angle is equal to the length of the side opposite the angle divided by the hypotenuse. So, we have:
sin(58°) = PQ / PR
We can rearrange that equation to solve for PQ:
PQ = sin(58°) * PR
Now we can substitute this value into the previous equation for QR^2:
QR^2 = (7.1 cm)^2 - (sin(58°) * PR)^2
Finally, we just need to find the square root of QR^2 to get the actual length of QR. And remember, there's always time for a quick joke: Why don't skeletons fight each other? They don't have the guts!
Now, let's calculate the area of △PQR using the formula: Area = 0.5 * PQ * QR. Once we plug in the values and calculate the area, we'll round it to the nearest tenth of a square centimeter.
I hope this explanation didn't make you triangulated!
To find the area of triangle PQR, we can use the formula:
Area = (1/2) * base * height.
In this case, the base of the triangle is PR, which is 7.1 cm.
To find the height of the triangle, we can use trigonometry. Since angle R is 90 degrees and angle P is 58 degrees, we can find the height using the sine function.
sin(P) = opposite/hypotenuse
sin(58) = height/7.1
Rearranging the equation:
height = sin(58) * 7.1
Now we can calculate the height using a calculator:
height = 0.848 * 7.1
height ≈ 6.0458 cm
Now we can substitute the values into the area formula:
Area = (1/2) * 7.1 * 6.0458
Area ≈ 21.4428 cm^2
Therefore, the area of triangle PQR is approximately 21.4 square centimetres to the nearest tenth.
To find the area of triangle PQR, we can use the formula for the area of a triangle based on its base and height. In this case, we need to find the height of triangle PQR.
Since ∠R = 90°, triangle PQR is a right-angled triangle, and the height will be the length of the line segment from vertex P to the side QR, perpendicular to QR.
To find this height, we can use the trigonometric function tangent. The tangent of an angle is equal to the ratio of the opposite side length to the adjacent side length.
In triangle PQR, we have:
tan(∠P) = Opposite/Adjecent
tan(58°) = QR/PR
To find the height QR, we rearrange the equation:
QR = tan(58°) * PR
Now, we can substitute the given values:
QR = tan(58°) * 7.1 cm
Using a scientific calculator, we can compute the value of tan(58°) to be approximately 1.619.
QR = 1.619 * 7.1 cm
QR ≈ 11.51 cm (rounded to the nearest hundredth)
Now that we have the height QR, we can use the formula for the area of a triangle:
Area = (1/2) * base * height
In this case, the base is PR and the height is QR:
Area = (1/2) * 7.1 cm * 11.51 cm
Using a calculator, we can find the area to be approximately 40.966 square centimeters.
Therefore, the area of triangle PQR is approximately 40.966 square centimeters, rounded to the nearest tenth of a square centimeter.