triangle GHE hypotenus GE=21,the altitude from H intersect GE @ F
GH=10
HE=17
a.Find HF
b.Is triangle EHF similar to triangle HGF ?
To find the length of HF, we need to use the properties of similar triangles.
a. Finding HF:
First, let's find the length of GF. Since GF is the remaining side of right triangle GHE with hypotenuse GE, we can use the Pythagorean theorem:
GF^2 = GE^2 - HE^2
GF^2 = 21^2 - 17^2
GF^2 = 441 - 289
GF^2 = 152
GF ≈ √152
GF ≈ 12.33
Now, let's find the length of GHF. Since H is the altitude from H to GE, GF and HE will form right triangle GHF:
GHF^2 = GH^2 - HF^2
10^2 = 12.33^2 - HF^2
100 = 152 - HF^2
HF^2 = 152 - 100
HF^2 = 52
HF ≈ √52
HF ≈ 7.21
Therefore, HF ≈ 7.21.
b. Determining similarity:
To determine if triangle EHF is similar to triangle HGF, we need to check if their corresponding angles are equal. Since angle EHF is a right angle, we need to compare it with angle HGF.
In triangle HGF, angle HGF is equal to the complement of angle EHF (90 degrees). Since the two angles are complementary, they are equal.
Therefore, triangle EHF is similar to triangle HGF.
b) should be answered first, then a) becomes trivial
b)yes, it is similar since there is a common angle at E and both have a 90º angle, so the third angle pair must be equal.
a) as a result, corresponding sides are in the same ratio.
HF/HE = HG/GE
HF/17 = 10/21
HF = 170/21