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Zorluk: OrtaPythagorean Theorem and Special Right Triangles

An architect is designing a triangular roof truss, △ABC\triangle ABC, where side ABAB is equal in length to side ACAC. The height of the truss, represented by the altitude from vertex AA to the base BCBC, is 1212 feet. If the measure of the base angle ∠ABC\angle ABC is 30∘30^\circ, what is the length, in feet, of the base BCBC? (Round your answer to the nearest tenth.)

Cevap: 41.6 feet

Cevap

The correct answer is 41.6
The correct answer is obtained by recognizing that the altitude of the isosceles triangle bisects the base into two congruent 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangles. The leg opposite the 30∘30^\circ angle is 1212 feet, so the leg adjacent (which is half the base) is 12312\sqrt{3} feet. Doubling this gives a total base length of 24324\sqrt{3} feet, which is approximately 41.641.6 feet when rounded to the nearest tenth.

Adım Adım Çözüm

1
Identify the right triangle formed by the altitude.
An altitude ADAD perpendicular to base BCBC, creating two right triangles, △ABD\triangle ABD and △ACD\triangle ACD, with AD=12AD = 12 feet.
In an isosceles triangle, the altitude to the base bisects the base and is perpendicular to it.
2
Determine the angles of the right triangle △ABD\triangle ABD.
Triangle △ABD\triangle ABD is a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ special right triangle.
Angle BB is 30∘30^\circ and angle ADBADB is 90∘90^\circ, leaving 60∘60^\circ for angle BADBAD.
3
Calculate the length of the segment BDBD.
BD=123BD = 12\sqrt{3} feet
In a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle, the longer leg is 3\sqrt{3} times the shorter leg (which is opposite the 30∘30^\circ angle).
4
Find the total length of the base BCBC.
BC=243BC = 24\sqrt{3} feet
Since DD is the midpoint of BCBC, the total length BCBC is 2×BD2 \times BD.
5
Convert the exact value to a decimal rounded to the nearest tenth.
BC≈41.6BC \approx 41.6
24×1.73205=41.56924 \times 1.73205 = 41.569, which rounds to 41.641.6.

Anahtar Kavram

Properties of special 30-60-90 right triangles and altitudes of isosceles triangles
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Konular

Pythagorean Theorem and Special Right Triangles

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