Examkin

Question

Difficulty: HardGeometric Figures on the Coordinate Plane

In the standard (x,y)(x,y) coordinate plane, a triangle has vertices at A(1,2)A(1, 2), B(9,2)B(9, 2), and C(5,8)C(5, 8). A horizontal line defined by the equation y=ky = k divides the area of the triangle into two regions of equal area. What is the value of kk?

  1. A
    55
  2. 8−328 - 3\sqrt{2}Answer
  3. C
    2+322 + 3\sqrt{2}
  4. D
    8−238 - 2\sqrt{3}
  5. E
    44

Answer

8−328 - 3\sqrt{2}
The correct answer is 8−328 - 3\sqrt{2}. Because the horizontal line y=ky=k is parallel to the base of the triangle, it creates a smaller top triangle similar to the original one. The area of the original triangle is 2424 and the area of the smaller triangle is 1212, yielding an area ratio of 12\frac{1}{2}. The height of the original triangle is 66, and the height of the smaller triangle is 8−k8-k. Since the ratio of the areas of similar triangles is the square of the ratio of their heights, we write 12=(8−k6)2\frac{1}{2} = \left(\frac{8-k}{6}\right)^2. Solving this equation yields k=8−32k = 8 - 3\sqrt{2}.

Step-by-Step Solution

1
Calculate the area of the entire triangle ABCABC.
The base AB‾\overline{AB} is horizontal along y=2y = 2 with a length of 9−1=89 - 1 = 8. The height is the vertical distance from y=2y = 2 to the vertex C(5,8)C(5, 8), which is 8−2=68 - 2 = 6. The area of △ABC\triangle ABC is 12×8×6=24\frac{1}{2} \times 8 \times 6 = 24.
Finding the total area is necessary to determine the target area of the divided regions.
2
Find the area of the smaller triangle formed above the line y=ky = k.
The line y=ky = k divides the triangle into a smaller top triangle and a bottom trapezoid. Since both regions have equal areas, the area of the smaller top triangle is 242=12\frac{24}{2} = 12.
The line divides the original area of 2424 into two equal parts of 1212 each.
3
Set up the area ratio equation using the properties of similar triangles.
The smaller top triangle is similar to △ABC\triangle ABC because its base is parallel to AB‾\overline{AB}. The ratio of their areas is the square of the ratio of their heights: 1224=(8−k6)2  ⟹  12=(8−k6)2\frac{12}{24} = \left(\frac{8 - k}{6}\right)^2 \implies \frac{1}{2} = \left(\frac{8 - k}{6}\right)^2.
For similar figures, the area ratio is equal to the square of the scale factor.
4
Solve the ratio equation for kk.
Taking the square root of both sides gives 12=8−k6\frac{1}{\sqrt{2}} = \frac{8 - k}{6}, which simplifies to 22=8−k6\frac{\sqrt{2}}{2} = \frac{8 - k}{6}. Multiplying both sides by 66 gives 32=8−k3\sqrt{2} = 8 - k, which yields k=8−32k = 8 - 3\sqrt{2}.
This isolates kk to find the exact vertical coordinate of the dividing line.

Key Concept

Using properties of similar figures to determine areas and coordinates on the coordinate plane.
Rate this question

Topics

Geometric Figures on the Coordinate Plane

More questions from this topic