GRE - Graduate Record Examinations Plane and Coordinate Geometry Questions and Answers 1 — Questions and Answers

Question 1: In the xy-plane, a triangle has vertices at A(1, 2), B(7, 2), and C(3, 8). What is the length of the median from vertex A to the side BC?

√41
5 (Correct answer)
6
√52

Correct answer: 5

The median from vertex A connects A to the midpoint of the opposite side, BC. First, find the midpoint M of BC using the midpoint formula: M = ((x₁+x₂)/2, (y₁+y₂)/2). M = ((7+3)/2, (2+8)/2) = (10/2, 10/2) = (5, 5). Next, find the length of the median AM using the distance formula: d = √((x₂-x₁)² + (y₂-y₁)²). The length of AM is √((5-1)² + (5-2)²) = √(4² + 3²) = √(16 + 9) = √25 = 5.

Question 2: Which of the following is an equation of a line that is perpendicular to the line 2x + 5y = 10 and passes through the point (4, 1)?

5x - 2y = 18 (Correct answer)
2x + 5y = 13
2x - 5y = 3
5x + 2y = 22

Correct answer: 5x - 2y = 18

First, find the slope of the given line, 2x + 5y = 10. Rearrange it into slope-intercept form (y = mx + b): 5y = -2x + 10, so y = (-2/5)x + 2. The slope (m₁) is -2/5. The slope of a perpendicular line (m₂) is the negative reciprocal of m₁, which is m₂ = 5/2. Now, use the point-slope form y - y₁ = m(x - x₁) with the point (4, 1) and slope 5/2: y - 1 = (5/2)(x - 4). To match the answer choices, convert this to standard form. Multiply by 2: 2(y - 1) = 5(x - 4). This simplifies to 2y - 2 = 5x - 20, and rearranging gives 5x - 2y = 18.

Question 3: What is the area of the quadrilateral with vertices at coordinates (-2, 1), (4, 1), (2, 5), and (0, 5) in the xy-plane?

20
24
16 (Correct answer)
32

Correct answer: 16

By plotting the vertices, we can identify the shape. The segment from (-2, 1) to (4, 1) is a horizontal line of length 6. The segment from (0, 5) to (2, 5) is a horizontal line of length 2. Since these two sides are parallel, the quadrilateral is a trapezoid. The parallel sides are the bases (b₁=6, b₂=2). The height (h) is the perpendicular distance between the bases, which is the difference in their y-coordinates: 5 - 1 = 4. The area of a trapezoid is A = (1/2)(b₁ + b₂)h. Plugging in the values: A = (1/2)(6 + 2)(4) = (1/2)(8)(4) = 16.

Question 4: A circle in the xy-plane has its center at (-3, 5). If the point (1, 8) lies on the circle, what is the area of the circle?

5π
25π (Correct answer)
10π
50π

Correct answer: 25π

The area of a circle is A = πr². The radius (r) is the distance from the center to any point on the circle. We can find the radius using the distance formula between the center (-3, 5) and the point (1, 8). r = √((1 - (-3))² + (8 - 5)²) = √((4)² + (3)²) = √(16 + 9) = √25 = 5. Now, we can calculate the area: A = π(5)² = 25π.

Question 5: Point P has coordinates (a, b). Point Q is the reflection of point P across the line y = x. Point R is the reflection of point Q across the y-axis. What are the coordinates of point R?

(b, -a)
(-a, b)
(a, -b)
(-b, a) (Correct answer)

Correct answer: (-b, a)

Let's follow the transformations step by step. The initial point is P(a, b). The rule for reflecting a point across the line y = x is to swap the coordinates. So, the coordinates of point Q are (b, a). The rule for reflecting a point across the y-axis is to negate the x-coordinate. So, reflecting Q(b, a) across the y-axis results in point R with coordinates (-b, a).

Question 6: What is the area of the triangle formed in the first quadrant by the line with the equation 3x + 4y = 24 and the coordinate axes?

12
24 (Correct answer)
48
18

Correct answer: 24

The vertices of the triangle are the origin (0,0), the x-intercept, and the y-intercept of the line. To find the x-intercept, set y=0 in the equation: 3x + 4(0) = 24, which gives 3x = 24, so x = 8. The x-intercept is (8, 0). To find the y-intercept, set x=0: 3(0) + 4y = 24, which gives 4y = 24, so y = 6. The y-intercept is (0, 6). The triangle is a right triangle with a base of 8 and a height of 6. The area is (1/2) * base * height = (1/2) * 8 * 6 = 24.

GRE - Graduate Record Examinations Plane and Coordinate Geometry Questions and Answers 1

In the xy-plane, a triangle has vertices at A(1, 2), B(7, 2), and C(3, 8).
What is the length of the median from vertex A to the side BC?