Given: PSTK is a rectangle
Area of PSTK = 562m^2
m ∠ TOK = 75 degrees
Find: PS, PK
If 56=2m+6,then m=
To find PS and PK, we need to analyze the properties of rectangles and use the given information.
1. Start with the formula for the area of a rectangle: Area = length × width. In this case, the area is given as 562 m^2.
2. Since PSTK is a rectangle, opposite sides are equal in length.
3. Let's assume PS to be the length and PK to be the width.
4. To find the lengths of PS and PK, we need to solve for PS and PK using the given area information.
5. Start by setting up an equation: PS × PK = Area. Substituting the given values, we get PS × PK = 562 m^2.
6. Now we need more information to solve for PS and PK. If we can find the length of PK or the length of one of the other sides, we can solve for the remaining side.
7. Since m ∠ TOK is given as 75 degrees, we can use this angle to find the length of PK.
8. Draw a line segment from P to meet side KT at a right angle. Label the point where the line segment intersects KT as Q.
9. Triangle PKQ is a right triangle with an angle at K equal to 75 degrees, and we know the area of the rectangle PSTK.
10. To find the length of PK, we can use trigonometry: PK = PT × sin(75°). However, we don't know the length of PT yet.
11. To find PT, we can use the area of PSTK and the width PK: PT = Area / PK. Substituting the given values, we get PT = 562 m^2 / PK.
12. Now we can substitute the value of PT in the equation for PK: PK = (562 m^2 / PK) × sin(75°).
13. Simplify the equation: PK^2 = 562 m^2 / sin(75°).
14. Solve for PK using a scientific calculator or online tool: PK ≈ 15.37 m (rounded to two decimal places).
15. Now that we have PK, we can substitute it back into the equation for PT to find the length of PT: PT = 562 m^2 / PK ≈ 36.57 m (rounded to two decimal places).
16. Therefore, PS = PT = 36.57 m.
So, PS ≈ 36.57 m and PK ≈ 15.37 m.
I assume O is the center of the rectangle. If so, then if the rectangle's dimensions are x and y, we have
xy = 562
x/y = tan(75/2)°
now solve for x and y