the lengths of the sides of a triangle are 10,17 and 21 cm. how long is the altitude of the triangle to the longest side?
sketch your triangle, I called BC = 21 , AB = 17
By the cosine law we can find angle B
then
sinB = altitude/17
altitude = 17sin B
To find the altitude of a triangle to the longest side, we can use the formula:
Altitude = (2 * Area) / Base
First, we need to find the area of the triangle. We can use Heron's formula to calculate it.
Step 1: Calculate the semi-perimeter (s) of the triangle by adding all the sides and dividing by 2.
s = (10 + 17 + 21) / 2 = 48 / 2 = 24 cm
Step 2: Calculate the area (A) of the triangle using Heron's formula.
A = √(s * (s - a) * (s - b) * (s - c))
where a, b, and c are the lengths of the sides of the triangle.
A = √(24 * (24 - 10) * (24 - 17) * (24 - 21))
= √(24 * 14 * 7 * 3)
= √(14112)
≈ 118.88 cm² (rounded to two decimal places)
Now, let's calculate the altitude.
Step 3: Find the length of the longest side, which is 21 cm.
Base = 21 cm
Step 4: Substitute the values into the formula to find the altitude.
Altitude = (2 * Area) / Base
= (2 * 118.88) / 21
≈ 5.68 cm (rounded to two decimal places)
Therefore, the altitude of the triangle to the longest side is approximately 5.68 cm.
To find the length of the altitude of a triangle to the longest side, we can use the formula:
Altitude = (2 * Area) / Base
First, we need to calculate the area of the triangle using Heron's formula:
s = (a + b + c) / 2
where s is the semi-perimeter and a, b, c are the lengths of the sides.
In this case, the lengths of the sides are 10, 17, and 21 cm. So,
s = (10 + 17 + 21) / 2
= 48 / 2
= 24 cm
Now, we can calculate the area of the triangle using Heron's formula:
Area = √(s * (s - a) * (s - b) * (s - c))
where a, b, c are the lengths of the sides.
Area = √(24 * (24 - 10) * (24 - 17) * (24 - 21))
= √(24 * 14 * 7 * 3)
= √(14112)
≈ 118.77 cm² (rounded to 2 decimal places)
Next, we can calculate the altitude of the triangle to the longest side:
Base = 21 cm (the length of the longest side)
Altitude = (2 * Area) / Base
= (2 * 118.77) / 21
≈ 5.66 cm
Therefore, the length of the altitude of the triangle to the longest side is approximately 5.66 cm.