Each base of an isosceles triangle measures 42 degrees, 30'. The base is 14.6 meters long.
a. Find the length of a leg of the triangle
b. Find the altitude of the triangle
c. What is the area of the triangle?
make a sketch, marking the angles
draw the altitude, cutting the base into 2 parts of 7.3 each, and giving you right-angled triangles.
a) leg of triangle = hypotenuse of the right-angled triangle, call it c
cos 42.5 = 7.3/c
c = 7.3/cos 42.5 = ....
b) altitude --- h
tan 42.5 = h/7.3
h = 7.3tan 42.5 = ....
c) area - (1/2) base x heigh
= (1/2)(14.6)(h) , the h from b)
= ....
a. Well, we can use some trigonometry to find the length of a leg of the triangle. Since each base angle measures 42 degrees, 30', we can split the triangle into two right triangles. Half of the base would be the adjacent side, and the length of a leg would be the opposite side. Using the trigonometric function tangent, we can say that tan(42 degrees, 30') = opposite / adjacent. So, the length of a leg would be (14.6 meters / 2) * tan(42 degrees, 30').
b. Now, to find the altitude of the triangle, we need to find the length of the perpendicular from the vertex to the base. Again, using trigonometry, we can say that sin(42 degrees, 30') = opposite / hypotenuse. The hypotenuse would be the length of a leg we found in part a. So, the altitude would be (length of a leg) * sin(42 degrees, 30').
c. Finally, to find the area of the triangle, we can use the formula A = (1/2) * base * height. The base is given as 14.6 meters, and we found the height in part b. So, we can substitute the values and calculate the area.
To find the length of a leg of the isosceles triangle and the altitude of the triangle, we can use the trigonometric relationships in a triangle.
a. Finding the length of a leg of the triangle:
The base angle of the isosceles triangle is 42 degrees, 30', which means each base angle is (1/2)*42 degrees, 30' = 21 degrees, 15'.
Using trigonometry, we can find the length of the leg using the tangent function:
tan(angle) = opposite/adjacent
tan(21 degrees, 15') = leg/ (1/2) * base
Let's calculate the length of the leg:
tan(21 degrees, 15') = leg / (1/2) * 14.6 meters
leg = tan(21 degrees, 15') * (1/2) * 14.6 meters
Calculating this, we find:
leg ≈ tan(21.25°) * 0.5 * 14.6 meters
leg ≈ 0.3853 * 0.5 * 14.6 meters
leg ≈ 2.832 meters
Therefore, the length of a leg of the isosceles triangle is approximately 2.832 meters.
b. Finding the altitude of the triangle:
The altitude of an isosceles triangle bisects the base and forms two right triangles. We can use the Pythagorean theorem to find the altitude.
Let h be the altitude of the triangle, then we have:
h^2 = (leg/2)^2 + base^2
Substituting the values we know:
h^2 = (2.832/2)^2 + 14.6^2
h^2 = 1.416^2 + 213.16
h^2 = 1.999856 + 213.16
h^2 = 215.159856
h ≈ √(215.159856)
h ≈ 14.660 meters (rounded to three decimal places)
Therefore, the altitude of the triangle is approximately 14.660 meters.
c. Finding the area of the triangle:
The area of a triangle can be calculated by multiplying the base by the altitude and dividing by 2.
Area = (base * altitude) / 2
Plugging in the values:
Area = (14.6 * 14.660) / 2
Area = 213.596
Therefore, the area of the triangle is 213.596 square meters.
To find the length of a leg of the triangle, you can use the law of cosines. The formula for the law of cosines is:
c^2 = a^2 + b^2 - 2ab * cos(C)
where a, b, and c represent the side lengths of the triangle, and C is the angle opposite to side c.
In this case, the base of the triangle is 14.6 meters long, and the angle at the base is 42 degrees, 30'. It is important to convert this angle into decimal form for calculations.
42 degrees, 30' = 42.5 degrees
Let's call the unknown length of the legs of the triangle "x".
Using the law of cosines, we have:
x^2 = 14.6^2 + 14.6^2 - 2 * 14.6 * 14.6 * cos(42.5)
Solving this equation will give you the length of one of the legs of the triangle.
To find the altitude of the triangle, you can use the formula for the area of a triangle:
Area = (1/2) * base * height
In this case, the base of the triangle is given as 14.6 meters. To find the height (altitude), you need to solve for it.
Using the formula, we have:
Area = (1/2) * 14.6 * height
Now, we need to know the area of the triangle in order to calculate the altitude.
To find the area of the triangle, you can use the formula:
Area = (1/2) * base * altitude
Since the base is given as 14.6 meters, we can solve for the altitude (height) using the given formula.
Finally, you will have the length of one leg of the triangle, the altitude, and the base, which is needed to find the area of the triangle.