The altitude drawn to the hypotenuse of a right triangle divides the hypotenuse into segments such that their lengths are in rat
io of 1:4. If the length of the altitude is 8, find the length of: (a) Each segment of the hypotenuse (b) The longer leg of the triangle.
1 answer:
In geometry, it would be always helpful to draw a diagram to illustrate the given problem.
This will also help to identify solutions, or discover missing information.
A figure is drawn for right triangle ABC, right-angled at B.
The altitude is drawn from the right-angled vertex B to the hypotenuse AC, dividing AC into two segments of length x and 4x.
We will be using the first two of the three metric relations of right triangles.
(1) BC^2=CD*CA (similarly, AB^2=AD*AC)
(2) BD^2=CD*DA
(3) CB*BA = BD*AC
Part (A)
From relation (2), we know that
BD^2=CD*DA
substitute values
8^2=x*(4x) => 4x^2=64, x^2=16, x=4
so CD=4, DA=4*4=16 (and AC=16+4=20)
Part (B)
Using relation (1)
AB^2=AD*AC
again, substitute values
AB^2=16*20=320=8^2*5
=>
AB
=sqrt(8^2*5)
=8sqrt(5)
=17.89 (approximately)
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Answer: x = 72
Step-by-step explanation: Although the problem is saying "X over 8", this really signifies that the <em>x</em> is being divided by 8 which is what a fraction is, numerator divided by a denominator.
Let's first undo what has been done. Since our variable is being divided by 8, we'll undo that by multiplying by 8 on both sides. It's important to remember, whatever you do on one side,
you <u><em>must</em></u> do to the other side.
If you look now, the 8 in the numerator and the 8 in the denominator will divide out and it will leave us with our variable which is <em>x</em>. Working on the other side now, 9 x 8 is equal to 72 so <em>x = 72</em> is our answer.
We can check our answer by plugging a 72 back into the original equation. So we have 72/8 = 9 which is a true statement
so our answer is correct.
