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In the xy-coordinate system, rectangle ABCD is inscribed within a circle having the equation x2 + y2 = 25. Line segment AC is a diagonal of the rectangle and lies on the x-axis. Vertex B lies in quadrant II and vertex D lies in quadrant IV. If side BC lies on line y = 3x + 15, what is the area of rectangle ABCD?
- a)15
- b)30
- c)40
- d)45
- e)50
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
In the xy-coordinate system, rectangle ABCD is inscribed within a circ...
The equation of a circle given in the form x2 + y2 = r2 indicates that the circle has a radius of r and that its center is at the origin (0,0) of the xy-coordinate system. Therefore, we know that the circle with the equation x2 + y2 = 25 will have a radius of 5 and its center at (0,0).
If a rectangle is inscribed in a circle, the diameter of the circle must be a diagonal of the rectangle (if you try inscribing a rectangle in a circle, you will see that it is impossible to do so unless the diagonal of the rectangle is the diameter of the circle). So diagonal AC of rectangle ABCD is the diameter of the circle and must have length 10 (remember, the radius of the circle is 5). It also cuts the rectangle into two right triangles of equal area. If we find the area of one of these triangles and multiply it by 2, we can find the area of the whole rectangle.
If a rectangle is inscribed in a circle, the diameter of the circle must be a diagonal of the rectangle (if you try inscribing a rectangle in a circle, you will see that it is impossible to do so unless the diagonal of the rectangle is the diameter of the circle). So diagonal AC of rectangle ABCD is the diameter of the circle and must have length 10 (remember, the radius of the circle is 5). It also cuts the rectangle into two right triangles of equal area. If we find the area of one of these triangles and multiply it by 2, we can find the area of the whole rectangle.
We could calculate the area of right triangle ABC if we had the base and height. We already know that the base of the triangle, AC, has length 10. So we need to find the height.
The height will be the distance from the x-axis to vertex B. We need to find the coordinate of point B in order to find the height. Since the circle intersects triangle ABCD at point B, the coordinates of point B will satisfy the equation of the circle x2 + y2 = 25. Point B also lies on the line y=3x+15, so the coordinates of point B will satisfy that equation as well.
Since the values of x and y are the same in both equations and since y=3x+15, we can substitute (3x + 15) for y in the equation x2 + y2 = 25 and solve for x:
So the two possible values of x are -4 and -5. Therefore, the two points where the circle and line intersect (points B and C) have x-coordinates -4 and -5, respectively. Since the x-coordinate of point C is -5 (it has coordinates (-5, 0)), the x-coordinate of point B must be -4. We can plug this into the equation and solve for the y-coordinate of point B:
y=3(-4) +15 →
y=-12+15→
y=3
So the coordinates of point B are (-4, 3) and the distance from the x-axis to point B is 3, making the height of triangle ABC equal to 3. We can now find the area of triangle ABC:
The area of rectangle ABCD will be twice the area of triangle ABC. So if the area of triangle ABC is 15, the area of rectangle ABCD is (2)(15) = 30.
Since the values of x and y are the same in both equations and since y=3x+15, we can substitute (3x + 15) for y in the equation x2 + y2 = 25 and solve for x:
So the two possible values of x are -4 and -5. Therefore, the two points where the circle and line intersect (points B and C) have x-coordinates -4 and -5, respectively. Since the x-coordinate of point C is -5 (it has coordinates (-5, 0)), the x-coordinate of point B must be -4. We can plug this into the equation and solve for the y-coordinate of point B:y=3(-4) +15 →
y=-12+15→
y=3
So the coordinates of point B are (-4, 3) and the distance from the x-axis to point B is 3, making the height of triangle ABC equal to 3. We can now find the area of triangle ABC:
The area of rectangle ABCD will be twice the area of triangle ABC. So if the area of triangle ABC is 15, the area of rectangle ABCD is (2)(15) = 30.
The correct answer is B
Most Upvoted Answer
In the xy-coordinate system, rectangle ABCD is inscribed within a circ...
To find the area of rectangle ABCD, we need to determine the length of its sides.
Step 1: Find the coordinates of points A, B, C, and D
Since line segment AC is a diagonal of the rectangle and lies on the x-axis, point A will have coordinates (0,0).
The equation of line BC is given as y = 3x - 15. To find the x-coordinate of point C, we substitute y = 0 into the equation:
0 = 3x - 15
3x = 15
x = 5
So point C has coordinates (5,0).
Since line segment AC is a diagonal of the rectangle and the rectangle is inscribed within the circle x^2 + y^2 = 25, we can find the coordinates of points B and D as follows:
For point B in quadrant II, the x-coordinate will be negative and the y-coordinate will be positive. Let's assume the x-coordinate of B is -a and the y-coordinate is b.
Since B lies on the circle, we have:
(-a)^2 + b^2 = 25
a^2 + b^2 = 25 (equation 1)
Since B lies on line BC, we can substitute the coordinates (-a,b) into the equation of line BC:
b = 3(-a) - 15
b = -3a - 15 (equation 2)
Solving equations 1 and 2 simultaneously will give us the coordinates of point B.
Similarly, for point D in quadrant IV, the x-coordinate will be positive and the y-coordinate will be negative. Let's assume the x-coordinate of D is d and the y-coordinate is -c.
We have:
d^2 + (-c)^2 = 25
d^2 + c^2 = 25 (equation 3)
Substituting the coordinates (d,-c) into the equation of line BC, we get:
-c = 3d - 15
c = -3d + 15 (equation 4)
Solving equations 3 and 4 simultaneously will give us the coordinates of point D.
Step 2: Find the length of side BC
Using the distance formula, the length of side BC can be found as:
√[(x2 - x1)^2 + (y2 - y1)^2]
Substituting the coordinates of points B and C into the formula, we get:
√[((-a) - 5)^2 + (b - 0)^2]
√[(a + 5)^2 + b^2]
Step 3: Find the area of the rectangle
The area of a rectangle is given by the formula: Area = length * width
In this case, the length is the distance between points A and C, which is 5 units. The width is the distance between points B and C, which we found in step 2.
So the area of rectangle ABCD is:
Area = 5 * √[(a + 5)^2 + b^2]
Step 4: Substitute the coordinates of points B and D into the area formula
Using the coordinates of points B and D that we found in step 1, we can substitute these values into the area formula to get the final
Step 1: Find the coordinates of points A, B, C, and D
Since line segment AC is a diagonal of the rectangle and lies on the x-axis, point A will have coordinates (0,0).
The equation of line BC is given as y = 3x - 15. To find the x-coordinate of point C, we substitute y = 0 into the equation:
0 = 3x - 15
3x = 15
x = 5
So point C has coordinates (5,0).
Since line segment AC is a diagonal of the rectangle and the rectangle is inscribed within the circle x^2 + y^2 = 25, we can find the coordinates of points B and D as follows:
For point B in quadrant II, the x-coordinate will be negative and the y-coordinate will be positive. Let's assume the x-coordinate of B is -a and the y-coordinate is b.
Since B lies on the circle, we have:
(-a)^2 + b^2 = 25
a^2 + b^2 = 25 (equation 1)
Since B lies on line BC, we can substitute the coordinates (-a,b) into the equation of line BC:
b = 3(-a) - 15
b = -3a - 15 (equation 2)
Solving equations 1 and 2 simultaneously will give us the coordinates of point B.
Similarly, for point D in quadrant IV, the x-coordinate will be positive and the y-coordinate will be negative. Let's assume the x-coordinate of D is d and the y-coordinate is -c.
We have:
d^2 + (-c)^2 = 25
d^2 + c^2 = 25 (equation 3)
Substituting the coordinates (d,-c) into the equation of line BC, we get:
-c = 3d - 15
c = -3d + 15 (equation 4)
Solving equations 3 and 4 simultaneously will give us the coordinates of point D.
Step 2: Find the length of side BC
Using the distance formula, the length of side BC can be found as:
√[(x2 - x1)^2 + (y2 - y1)^2]
Substituting the coordinates of points B and C into the formula, we get:
√[((-a) - 5)^2 + (b - 0)^2]
√[(a + 5)^2 + b^2]
Step 3: Find the area of the rectangle
The area of a rectangle is given by the formula: Area = length * width
In this case, the length is the distance between points A and C, which is 5 units. The width is the distance between points B and C, which we found in step 2.
So the area of rectangle ABCD is:
Area = 5 * √[(a + 5)^2 + b^2]
Step 4: Substitute the coordinates of points B and D into the area formula
Using the coordinates of points B and D that we found in step 1, we can substitute these values into the area formula to get the final
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In the xy-coordinate system, rectangle ABCD is inscribed within a circle having the equation x2 + y2 = 25. Line segment AC is a diagonal of the rectangle and lies on the x-axis. Vertex B lies in quadrant II and vertex D lies in quadrant IV. If side BC lies on line y = 3x + 15, what is the area of rectangle ABCD?a)15b)30c)40d)45e)50Correct answer is option 'B'. Can you explain this answer?
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In the xy-coordinate system, rectangle ABCD is inscribed within a circle having the equation x2 + y2 = 25. Line segment AC is a diagonal of the rectangle and lies on the x-axis. Vertex B lies in quadrant II and vertex D lies in quadrant IV. If side BC lies on line y = 3x + 15, what is the area of rectangle ABCD?a)15b)30c)40d)45e)50Correct answer is option 'B'. Can you explain this answer? for GMAT 2024 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about In the xy-coordinate system, rectangle ABCD is inscribed within a circle having the equation x2 + y2 = 25. Line segment AC is a diagonal of the rectangle and lies on the x-axis. Vertex B lies in quadrant II and vertex D lies in quadrant IV. If side BC lies on line y = 3x + 15, what is the area of rectangle ABCD?a)15b)30c)40d)45e)50Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for GMAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the xy-coordinate system, rectangle ABCD is inscribed within a circle having the equation x2 + y2 = 25. Line segment AC is a diagonal of the rectangle and lies on the x-axis. Vertex B lies in quadrant II and vertex D lies in quadrant IV. If side BC lies on line y = 3x + 15, what is the area of rectangle ABCD?a)15b)30c)40d)45e)50Correct answer is option 'B'. Can you explain this answer?.
In the xy-coordinate system, rectangle ABCD is inscribed within a circle having the equation x2 + y2 = 25. Line segment AC is a diagonal of the rectangle and lies on the x-axis. Vertex B lies in quadrant II and vertex D lies in quadrant IV. If side BC lies on line y = 3x + 15, what is the area of rectangle ABCD?a)15b)30c)40d)45e)50Correct answer is option 'B'. Can you explain this answer? for GMAT 2024 is part of GMAT preparation. The Question and answers have been prepared according to the GMAT exam syllabus. Information about In the xy-coordinate system, rectangle ABCD is inscribed within a circle having the equation x2 + y2 = 25. Line segment AC is a diagonal of the rectangle and lies on the x-axis. Vertex B lies in quadrant II and vertex D lies in quadrant IV. If side BC lies on line y = 3x + 15, what is the area of rectangle ABCD?a)15b)30c)40d)45e)50Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for GMAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In the xy-coordinate system, rectangle ABCD is inscribed within a circle having the equation x2 + y2 = 25. Line segment AC is a diagonal of the rectangle and lies on the x-axis. Vertex B lies in quadrant II and vertex D lies in quadrant IV. If side BC lies on line y = 3x + 15, what is the area of rectangle ABCD?a)15b)30c)40d)45e)50Correct answer is option 'B'. Can you explain this answer?.
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