Each base angle of an isosceles triangle measures 42°. The base is 14.6 feet long.
A) Find the length of a leg of the triangle. Round to the nearest tenth of a foot.
B) Find the altitude of the triangle. Round to the nearest tenth of a foot.
One leg, half the base, and the altitude form a right triangle. The angle opposite the altitude is 42 degrees. Draw a figure and see.
leg length = (L/2)/cos 42 = 9.82 foot (round to 9.8)
altitude = (L/2) tan 42 = 6.57 foot
(round to 6.6)
To find the length of a leg of the isosceles triangle, we can use the trigonometric ratios specific to this triangle.
A) The angle between one of the base angles and the base of the triangle is 180° - 2 * 42° = 96° (the sum of angles in a triangle is 180°).
To find the length of a leg, we can use the sine ratio.
sin(angle) = opposite / hypotenuse
In this case, the "opposite" side is the leg of the triangle, and the "hypotenuse" is the base.
sin(96°) = leg / 14.6
leg = 14.6 * sin(96°)
Using a calculator or table, we find:
leg ≈ 14.6 * 0.841
leg ≈ 12.27 feet
Therefore, the length of a leg of the triangle is approximately 12.3 feet (rounded to the nearest tenth of a foot).
B) To find the altitude of the isosceles triangle, we can divide it into two congruent right-angled triangles by drawing an altitude from the top vertex to the base.
Since we know the base length is 14.6 feet and the base angle is 42°, we have a right-angled triangle with one side (the base) and one angle (the base angle) known.
We can use the trigonometric ratios again, this time using the tangent ratio.
tan(angle) = opposite / adjacent
In this case, the "opposite" side is the altitude of the triangle, and the "adjacent" side is half of the base (base / 2).
tan(42°) = altitude / (14.6 / 2)
altitude = (14.6 / 2) * tan(42°)
Using a calculator or table, we find:
altitude ≈ 7.3 * tan(42°)
altitude ≈ 7.3 * 0.907
altitude ≈ 6.62 feet
Therefore, the altitude of the triangle is approximately 6.6 feet (rounded to the nearest tenth of a foot).