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anemone
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Find the lengths of the sides of a triangle with 14, 22 and 28 as the lengths of its altitude.
anemone said:Find the lengths of the sides of a triangle with 14, 22 and 28 as the lengths of its altitude.
kaliprasad said:let the lengths opposite to altitudes of lengths 14,22,28 be x,y,z
then 2 times area
$14x = 22 y = 28 z$
or $x:: y:: z = \frac{1}{14}::\frac{1}{22}:: \frac{1}{28} = 44::28:: 22$(note all kept even to avoid fraction in computation )
so let x= 44t, y = 28t, z = 22t
area = $\dfrac{14x}{2}= 7x = 7 * 44 t = 308t$
using heros formula ( we get $s= \frac{44+28+22}{2}=47$)
$area = \sqrt{47*(47-44)*(47-28)*(47-22)}t^2 = \sqrt{47* 3 * 19* 25}t^2$
or $area = 5\sqrt{47*57}= 5\sqrt{2679}t^2$
so $308t = 5 \sqrt{2679}t^2$
or $t=\frac{308}{5\sqrt{2679}}$
so the sides are
$44t,28t,22t$ where t is as above
The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. To find the length of a side, you can use the formula a^2 + b^2 = c^2, where a and b are the lengths of the other two sides, and c is the length of the hypotenuse.
The Law of Sines states that in a triangle, the ratio of the length of a side to the sine of its opposite angle is the same for all three sides. The formula for finding the length of a side is a/sinA = b/sinB = c/sinC, where a, b, and c are the lengths of the sides, and A, B, and C are the opposite angles.
The Law of Cosines states that in a triangle, the square of the length of one side is equal to the sum of the squares of the other two sides, minus twice the product of the lengths of those sides and the cosine of the angle between them. The formula for finding the length of a side is c^2 = a^2 + b^2 - 2abcosC, where c is the length of the side, and a, b, and C are the lengths of the other two sides and the angle between them.
In a right triangle, you can use the Pythagorean Theorem to find the length of a side, whereas in an oblique triangle, you need to use either the Law of Sines or the Law of Cosines. Additionally, in a right triangle, the side opposite the right angle is always the hypotenuse, whereas in an oblique triangle, any of the three sides can be the longest.
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. While this theorem can help determine if a triangle is possible or not, it cannot be used to find the length of a side.