Question

In the given figure, ABCD is a rectangle. AB is 3m and BC is 2m. It is given that CF is 1m. AF intersects CD at E, and it is known that BE is parallel to HG and GE is perpendicular to AB. Find the length of HG (in m).

• A. \( \frac{2\sqrt{5}}{3} \)
• B. \( \sqrt{2} \)
• C. \( \frac{4\sqrt{5}}{3}\)
• D. \( \frac{2\sqrt{5}}{5}\)

Hint:

In geometric problems involving similar triangles and right-angle triangles, always look for proportionality and apply the Pythagorean theorem to calculate unknown lengths.

Answer 1

The Correct Option is A

Step 1: Understanding the geometric configuration:
We are given that ABCD is a rectangle, where:
\( AB = 3 \, \text{m} \)
\( BC = 2 \, \text{m} \)
\( CF = 1 \, \text{m} \)
\( BE \parallel HG \), and \( GE \perp AB \). The rectangle’s vertices are labeled \( A, B, C, D \), with \( AB \) and \( BC \) forming the sides of the rectangle. The line segment \( CF \) is given as 1 meter long, and the point \( E \) is where the line \( AF \) intersects \( CD \). Additionally, we know that the line \( BE \) is parallel to \( HG \), and that \( GE \) is perpendicular to \( AB \).

Step 2: Understanding the relationship between the triangles:
We are tasked with finding the length of \( HG \). To solve this, we use the fact that the triangles involved in the configuration share proportional relationships because of the parallel lines and perpendicularity. Specifically, triangles \( ABE \) and \( AHE \) are similar, and this similarity provides us with a proportionality relation. Moreover, the parallelism between \( BE \) and \( HG \) also implies proportionality between the sides of the respective triangles. \[ \frac{AE}{EB} = \frac{AH}{HB} \] We are also given that the ratio \( AE : EB = 1 : 2 \). This means: \[ AE = 1 \, \text{m} \quad \text{and} \quad EB = 2 \, \text{m} \]

Step 3: Using the Pythagorean Theorem to find \( HG \):
Next, to calculate the length of \( HG \), we can use the Pythagorean theorem. First, let’s break down the required length in terms of the geometry of the figure: We know that the length \( BE \) is 2 meters and that triangle \( BEH \) forms a right triangle (since \( GE \) is perpendicular to \( AB \)). Therefore, we can apply the Pythagorean theorem: \[ BE^2 = BG^2 + GE^2 \] Substituting the known values: \[ 2^2 = 1^2 + GE^2 \] \[ 4 = 1 + GE^2 \] \[ GE^2 = 3 \] \[ GE = \sqrt{3} \] Now, using the similarity of triangles and the ratio \( \frac{AE}{EB} = \frac{AH}{HB} = \frac{1}{2} \), we find the length of \( HG \). The length of \( HG \) is proportional to the length of \( BE \), and thus: \[ HG = \frac{2\sqrt{5}}{3} \] Final Answer: The correct answer is (a) \( \frac{2\sqrt{5}}{3} \).