A triangle has side lengths 10,17 and 21cm. Find the length of the shortest altitude.
And please show working out to help me understand.
if a,b,c = 10,17,21,
a^2 = b^2+c^2-2bc cos C, so
cosC = (17^2 + 21^2 - 10^2)/(2*17*21) = 15/17
so, sinC = 0.47
figure sinA and sinB the same way
If the altitudes from A,B,C are Ha,Hb,Hc, then
Hc = a*sinB = b*sinA
Figure the other altitudes in like wise.
OR--------
figure the area using Heron's formula.
Then since 1/2 c*Hc = area
Hc = 2*area/c
Since the area does not change, the shortest altitude is on the longest side.
To find the length of the shortest altitude of a triangle, we can use Heron's formula to first calculate the area of the triangle. Then we can use the formula for the length of an altitude of a triangle, which is given by the formula:
Altitude = (2 * Area) / Base
Step 1: Calculate the semi-perimeter (s) of the triangle using the formula:
s = (a + b + c) / 2
where a, b, and c are the side lengths of the triangle.
In this case, the side lengths are: a = 10 cm, b = 17 cm, and c = 21 cm.
s = (10 + 17 + 21) / 2
s = 48 / 2
s = 24 cm
Step 2: Calculate the area (A) of the triangle using Heron's formula:
A = sqrt(s * (s - a) * (s - b) * (s - c))
where sqrt denotes the square root.
A = sqrt(24 * (24 - 10) * (24 - 17) * (24 - 21))
A = sqrt(24 * 14 * 7 * 3)
A = sqrt(56448)
A ≈ 237.57 cm² (rounded to two decimal places)
Step 3: Calculate the length of the shortest altitude (h) using the formula:
h = (2 * A) / c
where c is the base of the triangle.
In this case, the base c is 21 cm.
h = (2 * 237.57) / 21
h = 475.14 / 21
h ≈ 22.63 cm (rounded to two decimal places)
Therefore, the length of the shortest altitude of the triangle with side lengths 10 cm, 17 cm, and 21 cm is approximately 22.63 cm.