solve this question. ( 3 points, P,Q,R are on a horizontal plane. point Q is on a bearing of 120° and a distance of 80m from P. Point R is on a bearing of 045°, and a distance of 150m from point Q. ¡. calculate correct to 2 significant figure the distance of P from R. ¡¡. Find the perpendicular distance from P to R. Illustrate it with a well labelled diagram.
solve this question. ( 3 points, P,Q,R are on a horizontal plane. point Q is on a bearing of 120° and a distance of 80m from P. Point R is on a bearing of 045°, and a distance of 150m from point Q. ¡. calculate correct to 2 significant figure the distance of P from R. ¡¡. Find the perpendicular distance from P to R. Illustrate it with a well labelled diagram.
The diagram I need it
Draw a diagram.
(i) use the law of cosines. The distance
PR^2 = 80^2 + 150^2 - 2*80*150 cos105°
(ii) perpendicular to what? I suspect you want one of the altitudes of triangle PQR. From (i) you now have all 3 sides, so you can find all 3 angles using the law of sines.
Using cosine rule
I.) q²=150²+80²-
(2×150×80)cos(105)°
q²=190.66~191m
II) perpendicular distance=sin24°×190/sin21°
=170m
Well, I'd love to help you with that, but I'm more of a humor bot than a geometry bot. I'm afraid I might draw things a bit crooked or make the numbers go in circles. If you have any other non-math related questions, feel free to ask, and I'll be here to make you laugh!
To solve this question, we need to break it down into steps and use concepts from trigonometry.
Step 1: Draw a well-labelled diagram
Let's start by drawing a diagram representing the given information. We have three points, P, Q, and R, on a horizontal plane. Point Q is at a bearing of 120° and a distance of 80m from P. Point R is at a bearing of 045° and a distance of 150m from Q.
P ---- Q ---- R
Step 2: Calculate the coordinates of point Q
To calculate the coordinates of point Q, we need to use trigonometry. From the given information, we know that the bearing of 120° means it forms an angle of 120° with the horizontal.
Using trigonometry, we can find the horizontal and vertical distances of point Q from point P using the given distance of 80m.
The horizontal distance can be found using the formula:
Horizontal distance = Distance × cos(angle)
Horizontal distance of Q = 80m × cos(120°) = -40m (we get a negative value because the angle is clockwise from the reference point)
Similarly, the vertical distance of Q can be found using the formula:
Vertical distance = Distance × sin(angle)
Vertical distance of Q = 80m × sin(120°) = 69.28m (rounded to two significant figures)
So, the coordinates of point Q are (-40m, 69.28m) with respect to point P.
Step 3: Calculate the coordinates of point R
To calculate the coordinates of point R, we can use the same approach as step 2. From the given information, we know that the bearing of 045° means it forms an angle of 45° with the horizontal.
The horizontal distance of R from Q can be found using the formula:
Horizontal distance = Distance × cos(angle)
Horizontal distance of R = 150m × cos(45°) = 106.07m (rounded to two significant figures)
Similarly, the vertical distance of R from Q can be found using the formula:
Vertical distance = Distance × sin(angle)
Vertical distance of R = 150m × sin(45°) = 106.07m (rounded to two significant figures)
Now, to get the coordinates of point R with respect to point P, we add the coordinates of Q to the distances obtained above:
Coordinates of R = (-40m + 106.07m, 69.28m + 106.07m) = (66.07m, 175.35m) (rounded to two significant figures).
So, the coordinates of point R are (66.07m, 175.35m) with respect to point P.
Step 4: Calculate the distance from P to R
We can now calculate the distance between points P and R using the coordinates obtained in step 3. We can use the distance formula:
Distance = √[(x2 - x1)^2 + (y2 - y1)^2]
Distance P to R = √[(66.07m - 0m)^2 + (175.35m - 0m)^2] = √[(4362.84) + (30711.92)] ≈ √35074.76 ≈ 187.14m (rounded to two significant figures).
Therefore, the distance from P to R is approximately 187.14m.
Step 5: Calculate the perpendicular distance from P to R
To find the perpendicular distance from P to R, we can use the vertical distance between the two points, which can be obtained from the coordinates calculated in step 3.
Perpendicular distance from P to R = 175.35m (rounded to two significant figures).
So, the perpendicular distance from P to R is approximately 175.35m.
Illustration:
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