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# If AB=20 and BC=25, what is the length of AD in the figure above?

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If AB=20 and BC=25, what is the length of AD in the figure above? [#permalink]

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01 Aug 2017, 12:21
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If AB=20 and BC=25, what is the length of AD in the figure above?

A. 6
B. 9.6
C. 12
D. 20
E. 24

[Reveal] Spoiler:
Attachment:

T6019.png [ 4.76 KiB | Viewed 956 times ]
[Reveal] Spoiler: OA

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Re: If AB=20 and BC=25, what is the length of AD in the figure above? [#permalink]

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01 Aug 2017, 13:00

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If AB=20 and BC=25, what is the length of AD in the figure above? [#permalink]

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01 Aug 2017, 19:21
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Given data:AB = 20, BC = 25

Since the triangle is a right angled triangle,
Using Pythagoras theorem
$$BC^2 = AB^2 + AC^2$$ => $$AC^2 = BC^2 - AB^2$$
$$AC^2 = 625 - 400$$
$$AC = \sqrt{225} = 15$$

The area of the triangle is 1/2 * AC * BD
The area if also equal to 1/2 * BC * AD

$$20*15 = 25*x$$

Solving for x, $$x = \frac{20*15}{25}$$= 12(Option C)
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If AB=20 and BC=25, what is the length of AD in the figure above? [#permalink]

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02 Aug 2017, 07:10
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Bunuel wrote:

If AB=20 and BC=25, what is the length of AD in the figure above?

A. 6
B. 9.6
C. 12
D. 20
E. 24
[Reveal] Spoiler:
Attachment:
T6019.png

These triangles (ABC and DAB) are similar 3-4-5 right triangles.

Both have one 90° angle, and angle B is common to both. If two angles are equal, the third is equal. (AA = AAA = similar triangles.)

Whenever a right triangle has longer leg length that is a multiple of 4 (20), and a hypotenuse that is a multiple of 5 (25), the figure is a 3-4-5 right triangle.

Or find AC, and the 3x-4x-5x ratio will be clear. Let AC = a, AB = b, and BC = c. By Pythagorean theorem, $$a^2 + b^2 = c^2$$

$$a^2 + 20^2 = 25^2$$
$$a^2 + 400 = 625$$
$$a^2 = 225$$
$$a = 15$$ = AC

15-20-25 divided by 5 throughout = 3-4-5 side ratio

Smaller triangle DAB must also be a 3-4-5 right triangle. Its hypotenuse AB = 20. To find length of shorter leg AD, find multiplier, using hypotenuse AB. 3x: 4x: 5x --> 5x is hypotenuse, 3x is shorter leg.

5x = 20
x = 4 = multiplier
AD corresponds with AC. Both are the shorter leg, i.e., 3x.

3x = (3)(4) = 12

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If AB=20 and BC=25, what is the length of AD in the figure above?   [#permalink] 02 Aug 2017, 07:10
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