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Zorluk: OrtaTriangles: Properties, Perimeter, and Area

An isosceles triangle has two sides of length 1010 units each and a base of length 1212 units. A line segment parallel to the base cuts through the triangle, creating a smaller top triangle with an area of 1212 square units. What is the perimeter of the resulting trapezoid?

Cevap: 28

Cevap

The perimeter of the trapezoid is 28.
First, find the height of the original isosceles triangle with sides 10, 10, and base 12 by applying the Pythagorean theorem to half of the base: h=102−62=8h = \sqrt{10^2 - 6^2} = 8. The area of the original triangle is 12×12×8=48\frac{1}{2} \times 12 \times 8 = 48 square units. Because the segment is parallel to the base, the smaller top triangle is similar to the original triangle. The ratio of their areas is 1248=14\frac{12}{48} = \frac{1}{4}, which means the linear scale factor is 14=12\sqrt{\frac{1}{4}} = \frac{1}{2}. Thus, the top triangle has legs of length 55 and a base of length 66. The remaining non-parallel sides of the trapezoid each measure 10−5=510 - 5 = 5 units, and its bottom base is 1212 units. Summing these four side lengths gives 5+5+6+12=285 + 5 + 6 + 12 = 28.

Adım Adım Çözüm

1
Calculate the height and area of the original isosceles triangle
The altitude to the base bisects the base into two segments of length 66. The altitude length is h=102−62=8h = \sqrt{10^2 - 6^2} = 8. The area of the original triangle is 12×12×8=48\frac{1}{2} \times 12 \times 8 = 48 square units.
Splitting the isosceles triangle along its altitude creates two right triangles with hypotenuse 10 and base leg 6.
2
Determine the linear scale factor of the smaller top triangle
The ratio of the area of the smaller triangle to the original triangle is 1248=14\frac{12}{48} = \frac{1}{4}. Taking the square root yields a linear scale factor of k=14=12k = \sqrt{\frac{1}{4}} = \frac{1}{2}.
A line parallel to the base forms a smaller triangle similar to the original triangle, and the ratio of areas of similar triangles is equal to the square of their linear scale factor.
3
Find the side lengths of the smaller triangle and the remaining side segments
The sides of the smaller triangle are 12×10=5\frac{1}{2} \times 10 = 5, 12×10=5\frac{1}{2} \times 10 = 5, and base 12×12=6\frac{1}{2} \times 12 = 6. The non-parallel side segments of the trapezoid are each 10−5=510 - 5 = 5.
Multiplying the dimensions of the original triangle by the linear scale factor gives the side lengths of the top triangle.
4
Calculate the perimeter of the trapezoid
Perimeter = 5+5+6+12=285 + 5 + 6 + 12 = 28.
Sum the lengths of the four boundary segments of the trapezoid.

Anahtar Kavram

Properties of isosceles triangles, Pythagorean theorem, area calculations, and similar triangle area ratios
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Konular

Triangles: Properties, Perimeter, and Area

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