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The circle c is enclosed in the square PQRS and touches all four sides, as shown in the diagram - Leaving Cert Mathematics - Question 4 - 2019
Question 4
The circle c is enclosed in the square PQRS and touches all four sides, as shown in the diagram. The co-ordinates of three of the vertices are P(2, 3), R(4, 17), and... show full transcript
Worked Solution & Example Answer:The circle c is enclosed in the square PQRS and touches all four sides, as shown in the diagram - Leaving Cert Mathematics - Question 4 - 2019
Step 1
Find the co-ordinates of Q.
Answer
To find the coordinates of point Q, we observe that the square's diagonal vertices are opposite. Given the vertices P(2, 3), R(4, 17), and S(-4, 11), we can determine the coordinates of Q by applying the properties of squares.
The coordinates of Q can be derived by averaging the x-coordinates and y-coordinates of the opposite corners:
- For x-coordinate:
[ x_Q = x_P + (x_R - x_S) = 2 + (4 - (-4)) = 2 + 8 = 10 ] - For y-coordinate:
[ y_Q = y_R + (y_S - y_P) = 17 + (-11 - 3) = 17 - 14 = 3 ]
Thus, the coordinates of Q are (10, 3).
Step 2
Find the co-ordinates of the centre of c.
Answer
The centre of circle c, which is also the centre of square PQRS, can be calculated by taking the average of the coordinates of points P, R, and S.
Using the formula for the midpoint: [ x_c = \frac{x_1 + x_2 + x_3}{3}, \quad y_c = \frac{y_1 + y_2 + y_3}{3} ] Substituting the vertices:
- For x-coordinate:
[ x_c = \frac{2 + 4 - 4}{3} = \frac{2}{3} \approx 0.67 ] - For y-coordinate:
[ y_c = \frac{3 + 17 + (-11)}{3} = \frac{9}{3} = 3 ]
Thus, the centre of c is approximately (0.67, 3).
Step 3
Find the length of the radius of c.
Answer
To find the radius of circle c, we can calculate the distance from the centre to the vertex P(2, 3). The radius can be found using the distance formula: [ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ] Substituting the coordinates of the centre (0.67, 3) and P(2, 3): [ r = \sqrt{(2 - 0.67)^2 + (3 - 3)^2} = \sqrt{(1.33)^2} = 1.33 ]
Thus, the radius of c is approximately 1.33 units.
Step 4
Find the equation of circle c.
Answer
The general equation of a circle is given by: [ (x - h)^2 + (y - k)^2 = r^2 ] where (h, k) is the centre and r is the radius. Substituting the values obtained:
- Centre (0.67, 3)
- Radius ( r = 1.33 )
Thus, the equation becomes: [ (x - 0.67)^2 + (y - 3)^2 = (1.33)^2 ] [ (x - 0.67)^2 + (y - 3)^2 = 1.77 ]
This represents the equation of circle c.