Question

\(A B C D\) is a rectangle of dimensions \(6 \mathrm{~cm} \times 8 \mathrm{~cm} . D E\) and \(B F\) are the perpendiculars drawn on the diagonal of the rectangle. What is the ratio of the shaded to that of unshaded region? (a) \(7: 3\) (b) \(16: 9\) (c) \(4: 3 \sqrt{2}\) (d) data insufficient

Short answer

Answer: The ratio of the shaded region to the unshaded region is 7:16.

Step-by-step solution

Find the Diagonal

First, we need to find the length of the diagonal in rectangle ABCD. By using the Pythagorean theorem, we can find the length of the diagonal (AC) as follows: \(AC^2 = AB^2 + BC^2\) \(AC^2 = 6^2 + 8^2\) \(AC^2 = 36 + 64 = 100\) \(AC = 10\)

Find the Area of Triangle DEE

To find the area of triangle DEE, we can use the fact that, \(Area_{DEE} = \frac{1}{2} DE \times EE\) Since triangle DEC is similar to triangle ABC (right-angled triangles with a common angle CED), we can use proportional lengths to find DE: \(\frac{DE}{6} = \frac{10}{8} \Rightarrow DE = \frac{15}{2}\) Now, we can find the area of triangle DEE: \(Area_{DEE} = \frac{1}{2} \frac{15}{2} \times \frac{15}{2} = \frac{225}{4}\)

Find the Area of Triangle BFF

Similarly, we can find the area of triangle BFF by using the proportionality and similarity of triangle ABC and BFC (right-angled triangles with a common angle FBG): \(\frac{BF}{8} = \frac{10}{6} \Rightarrow BF = \frac{40}{3}\) Now, we can find the area of triangle BFF: \(Area_{BFF} = \frac{1}{2} \frac{40}{3} \times \frac{40}{3} = \frac{800}{9}\)

Calculate the Area of the Shaded Region

Now we can find the area of the shaded region by subtracting the area of the triangles DEE and BFF from the area of the rectangle ABCD: \(Area_{Shaded} = Area_{ABCD} - Area_{DEE} - Area_{BFF}\) \(Area_{Shaded} = (6 \times 8) - \frac{225}{4} - \frac{800}{9}\) \(Area_{Shaded} = 48 - \frac{225}{4} - \frac{800}{9}\) \(Area_{Shaded} = \frac{192 - 225 - 320}{36}\) \(Area_{Shaded} = \frac{527}{36}\)

Calculate the Ratio of Shaded to Unshaded Regions

To find the ratio of shaded to unshaded regions, we need to find the area of the unshaded region. \(Area_{Unshaded} = Area_{DEE} + Area_{BFF}\) \(Area_{Unshaded} = \frac{225}{4} + \frac{800}{9}\) Finally, to find the ratio: \(\frac{Area_{Shaded}}{Area_{Unshaded}} = \frac{\frac{527}{36}}{\frac{225}{4} + \frac{800}{9}}\) \(\frac{Area_{Shaded}}{Area_{Unshaded}} = \frac{\frac{527}{36}}{\frac{1574}{36}}\) \(\frac{Area_{Shaded}}{Area_{Unshaded}} = \frac{527}{1574}\) After simplification, we get the ratio: \(\frac{Area_{Shaded}}{Area_{Unshaded}} = \frac{7}{16}\) This means that the ratio of the shaded region to the unshaded region is \(\boxed{7:16}\).

Key concepts

Pythagorean Theorem

Understanding the Pythagorean theorem is crucial when solving geometry problems involving right-angled triangles. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

This theorem is elegantly written as \( c^2 = a^2 + b^2 \) where \( c \) is the hypotenuse while \( a \) and \( b \) are the other two sides. For example, to find the diagonal of a rectangle - which is actually the hypotenuse of the right-angled triangles formed by the rectangle's sides - this theorem is essential. In the provided problem, the theorem helped solve for the diagonal of the rectangle, leading to further calculations to find the area of the shaded region.

Here's how it worked: the sides of the rectangle were \( 6 \) cm and \( 8 \) cm, so applying the Pythagorean theorem, we calculated \( AC^2 = 6^2 + 8^2 \) which gives us \( AC = 10 \) cm. Precise use of the Pythagorean theorem is the foundation to progress through various stages of geometrical problem solving.

Area Calculation

Area calculation is a fundamental aspect of geometry that helps in determining the size of two-dimensional shapes. For rectangles, the area is calculated by multiplying the length by the width. In our rectangle \( ABCD \), we calculate its area as \( 6 \) cm times \( 8 \) cm, resulting in \( 48 \) square centimeters.

However, it gets more interesting when you need to find the area of triangles, especially when they are not standard shapes. The generic formula for a triangle’s area is \( \frac{1}{2} \times base \times height \). In complex problems - like those involving diagonal divisions or other non-standard shapes within a rectangle - we have to use the property of similar triangles to determine the corresponding heights and bases, as seen with triangles \( DEC \) and \( BFC \) in this exercise.

Once the dimensions are known, we can compute the areas of triangles \( DEE \) and \( BFF \) using the formula. As exemplified, the understanding of how to apply the area formula flexibly is key to solving intricate geometry problems.

Similar Triangles

Similar triangles play a pivotal role in solving many geometry problems because they have corresponding angles that are equal and their sides have a constant ratio. When two triangles are similar, we can find the length of an unknown side if we know the scale factor (the ratio between the lengths of two corresponding sides).

In the given problem, triangles \( DEC \) and \( ABC \) are similar, as are triangles \( BFC \) and \( ABC \) because they share common angles and their sides are proportional. The relationships between the sides of the similar triangles allow us to calculate the necessary lengths to find the areas of the shaded and unshaded regions.

For instance, to find DE, we set up the proportion \( \frac{DE}{AB} = \frac{AC}{BC} \) and solve for DE. Similar steps are taken to find BF. Recognizing and correctly applying the concept of similar triangles can make solving complex geometric problems, which might first appear intractable, much more manageable.