triangle GHE
GH=10
HE = 17
hypotenuse GE =21
HF is an altitude
Find HF
10, 17, 21 IS NOT a Right triangle, no hypotenuse here
In general
h = 2*area/base
here the base is EG = 21
the area is
sqrt[ s(s-a)(s-b)(s-c) ]
where s = (a+b+c)/2
To find the length of HF, we can use the Pythagorean theorem since triangle GHE is a right triangle.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
In this case, GH and HE are the two legs of the triangle, and GE is the hypotenuse.
So, using the Pythagorean theorem, we have:
GH^2 + HF^2 = GE^2
Substituting the given values:
10^2 + HF^2 = 21^2
Simplifying:
100 + HF^2 = 441
Subtracting 100 from both sides:
HF^2 = 341
To find HF, we need to take the square root of both sides:
√(HF^2) = √341
Therefore, HF ≈ 18.46 (rounded to two decimal places).
So the length of HF is approximately 18.46 units.
To find the length of HF, we can use the Pythagorean Theorem. According to the theorem, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, triangle GHE is a right triangle because HF is an altitude, which means it is perpendicular to the base GE. Therefore, we can use the Pythagorean theorem as follows:
GH^2 + HE^2 = GE^2
Substituting the given values:
10^2 + 17^2 = 21^2
Simplifying:
100 + 289 = 441
389 = 441
Now, to find HF, we can use the fact that HF is the altitude from vertex H to the line GE. Since the altitude is perpendicular to the base, it creates two right triangles, namely triangle HGF and triangle HFE.
We know GH = 10, and GF is a part of the hypotenuse, so we need to find GF. Since GE is the hypotenuse and HF is the altitude, we can use the property that the lengths GH and HE are divided proportionally by the altitude HF. This means that:
GH/HF = HE/GF
Substituting the given values:
10/HF = 17/GF
Now, we can solve for GF by cross-multiplying:
10 * GF = 17 * HF
GF = (17 * HF) / 10
Now, we know that GF is a part of the hypotenuse GE, so GF + HF = GE. Substituting the values:
(17 * HF) / 10 + HF = 21
Simplifying:
(17 * HF + 10 * HF) / 10 = 21
27 * HF / 10 = 21
Now, to solve for HF, we can cross-multiply:
27 * HF = 21 * 10
HF = (21 * 10) / 27
Simplifying:
HF = 210 / 27
HF ≈ 7.77 (rounded to two decimal places)
Therefore, the length of HF is approximately 7.77.