If IJ=16 and JK=9, what is the length of HK⎯⎯⎯⎯⎯⎯?
The figure shows right triangle K H I with altitude H J to hypotenuse K I.
A. HK = 20
B. HK = 15
C. HK = 25
D. HK = 12
please help explain this to me thank you.
@oobleck how are those similar one is straight
what do I need to do then
To find the length of HK, we need to use the relationship between similar triangles and the Pythagorean theorem.
First, let's analyze the given right triangle KHI with altitude HJ.
Since HJ is an altitude, it forms two smaller right triangles: IJH and JHK.
Using the Pythagorean theorem, we have:
(IH)^2 + (HJ)^2 = (IJ)^2
(IH)^2 + (9)^2 = (16)^2
(IH)^2 + 81 = 256
(IH)^2 = 256 - 81
(IH)^2 = 175
Now, let's find the length of HK.
Since triangle KHI and triangle HKJ share the same angle at H, they are similar triangles.
According to the similarity of triangles KHI and HKJ, we can set up the following proportion to find the length of HK:
(HK) / (KJ) = (HI) / (IJ)
Substituting the given values, we have:
(HK) / 9 = (IH) / 16
Since we found that (IH)^2 = 175, we can substitute this value:
(HK) / 9 = sqrt(175) / 16
To isolate HK, multiply both sides of the equation by 9:
HK = (9 * sqrt(175)) / 16
Now, we can simplify the expression by finding the square root of 175:
HK = 9 * 13.23 / 16
HK ≈ 11.95
The length of HK is approximately 11.95.
Comparing the options given, we can see that the closest value to 11.95 is 12.
So the correct answer is D. HK = 12.
triangles IJK and KHJ are similar
see what you can do with that.