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Flura [38]
3 years ago
11

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.

Mathematics
1 answer:
Zanzabum3 years ago
6 0

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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