Town P is on bearing 315 degree from town Q. While town R is south of town P and west of town Q. If town R is 60km away from Q, how fara is R from P?
360 - 315 = 45
so P is 45 degrees north of west from Q
QRP is right angle
so PR = RQ = 60 km
The diagram shows a right angled
360-315=45
SOHCAHTOA
Tan45=x/60
Note: tan45=1
1=x/60
Cross multiply
60*1=x
X=60
Well, well, well! Looks like town R is playing hide and seek with town P and Q! Let's use our compass and a pinch of geometry to solve this riddle.
Town P is on a bearing of 315 degrees from town Q. If we follow that bearing, we end up going northwest. Now, town R is not far behind and is south of P and west of Q.
Since town R is 60 km away from Q, we can draw a right-angled triangle with Q, R, and P. The side opposite the right angle is 60 km (Q to R), and we want to find the hypotenuse (P to R).
Now, brace yourself for a comedy twist: we have ourselves a lovely 45-45-90 right triangle! The distance from Q to P will be equal, making it a 45-degree angle.
In this type of triangle, the hypotenuse is √2 times longer than the leg. So, if the distance from Q to R is 60 km, the distance from P to R will be 60 km times √2.
Calculating that, we find the distance from P to R will be approximately 84.85 km. Voila! Town R is about 84.85 km away from town P.
Hope that answers your question with a hint of humor!
To find the distance between town R and town P, we need to use trigonometry and the information given.
First, let's visualize the situation. Place town Q at the origin (0,0) on a coordinate system. Since town R is located 60km south of Q, its position can be represented as (0,-60).
Now, we know that town P is on a bearing of 315 degrees from town Q. A bearing is the direction or angle measured in degrees clockwise from the North direction.
To determine the position of town P, we can start by finding the bearing from Q to P. Since the bearing is 315 degrees, we need to go 45 degrees west from the North direction (360 - 315 = 45). Therefore, the bearing from Q to P is 45 degrees.
Next, we can find the coordinates of town P using trigonometry. Since the distance between Q and P is unknown, we'll represent it as 'x'. Using trigonometric ratios, we can determine the values of x and y coordinates of town P.
The trigonometric ratios we'll be using are:
sin(theta) = opposite/hypotenuse
cos(theta) = adjacent/hypotenuse
In our case:
sin(45) = y / x
cos(45) = x / x
Simplifying the equations:
sin(45) = y / x
cos(45) = 1
Since cos(45) = 1/√2 = x / x, we can simplify it to:
1/√2 = 1
Now, we have:
sin(45) = y / x
1/√2 = 1
From sin(45) = y / x, we can rewrite it as:
y = x * sin(45)
Since 1/√2 = 1, it implies that y = x.
So, we have:
y = x
Now, let's substitute y with -60 and solve for x:
-60 = x
Therefore, x = -60.
Now, we have the coordinates of town P as (x, y) = (-60, -60).
To find the distance between town P and town R, we can use the distance formula:
Distance = √ [(x2 - x1)^2 + (y2 - y1)^2]
Substituting the values into the equation:
Distance = √ [(-60 - 0)^2 + (-60 - (-60))^2]
Distance = √ [(-60)^2 + (0)^2]
Distance = √ [3600]
Distance = 60 km
Therefore, town R is 60 km away from town P.
My diagram shows a right-angled triangle PQR, with angle Q as 45° and RP = 60
So you have a right-angled isosceles triangle.
No calculations required, RQ is also 60 km