Question

The base of a triangular piece of paper ABC is 12 cm long. The paper is folded down over the base, with the crease DE parallel to the base of the paper. The area of the triangle that projects below the base is 16% that of the area of the triangle

ABC. The length of DE, in cm, is
(A) 9.6 (B) 8.4 (C) 7.2
(D) 4.8 (E) 6.96

Answer 1

The length of the crease DE in the paper triangle is 4.8 cm, which is determined by using the ratio of the areas of the triangles ABC and DE, and knowing that the area of the projecting triangle is 16% that of triangle ABC.

Step-by-step explanation:

To find the length of the crease DE, we start by understanding that DE is parallel to the base AB, and thus creates a similar smaller triangle within triangle ABC. Since the area of the projecting triangle is 16% that of triangle ABC, we can set up a ratio to solve for the unknown length of DE.

The area of a triangle is calculated using the formula ½ × base × height. If the crease DE is x cm from the base AB, then the areas of the similar triangles are proportional, and the ratio of their bases is the square root of the ratio of their areas. In this case, the ratio of the areas is 16:100 or 4:25 after reduction, which simplifies to 2:5 when we take the square root for the ratio of the bases (since the height remains the same for both triangles).

Thus, if AB = 12 cm, we have AB/DE = 5/2, and DE = (2/5) × 12 cm = 4.8 cm. Therefore, the correct answer is (D).

Answer 2

The length of DE, the crease formed when a triangular piece of paper is folded over its base, is approximately 1.92 cm.

Step-by-step explanation:

To find the length of DE, we need to determine the area below the base. Let's denote the height of the triangle ABC as h. The area of the triangle is given by the formula: Area = (1/2) * base * height. If the area of the section below the base is 16% of the total area, then the area below the base is (16/100) * (1/2) * 12 * h = 0.08 * 12 * h = 0.96h. We know that this area is equal to (1/2) * 12 * DE, so: 0.96h = 6DE. Therefore, DE = 0.96h/6 = 0.16h.

Since the height of the triangle ABC is not given, we cannot determine the exact length of DE. However, we can still solve the problem by finding a relationship between DE and the height using similar triangles. If we consider the triangle ABC and the triangle ADE, they share the same base DE and have proportional heights, h and x respectively. Therefore, we can set up the following proportion: h/DE = x/12. Plugging in the value of DE = 0.16h, we get: h/(0.16h) = x/12. Solving for x, we find: x = (0.16h * 12)/h = 1.92.

Therefore, the length of DE is approximately 1.92 cm.