Question

Question #72637

Answer 1

The first triangle has dimensions of 3 ft, 7.2 ft, and 7.8 ft.
The second triangle has dimensions of 9 ft, 21.6 ft, and 23.4 ft.

Explanation:

Information
we must have a ratio of `3:1` between all comparison pairs.

• first leg first triangle: 3 ft. second triangle: `x_1` ft.
• hypotenuse first triangle: `x_2` ft. second triangle: 23.4 ft.

The second leg will be addressed after we solve the first two sets.
also, both ramps form right triangles.

Since the hypotenuse of the second triangle is very much larger than the leg of the first, the second triangle is likely the bigger one.

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Finding the first leg: second triangle
`x_1/3=3/1`
`x_1=3\cdot3=9` ft

Finding the hypotenuse: first triangle
`23.4/x_2=3/1`
`3x_2=23.4`
`x_2=7.8` ft

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Apply Pythagoras' Theorem (`a^2+b^2=c^2`) to one of the triangles in order to find the remaining, second leg. Then use the `3:1` ratio to find the corresponding side on the other triangle.

Let's use the first triangle for convenience:
`3^2+b^2=7.8^2`
`9+b^2=60.84`
`b=\sqrt{51.84}=7.2` ft

Now combine with the ratio to find the second triangle's remaining leg. We will call the leg `\beta`
`\beta/7.2=3/1`
`\beta=3(7.2)=21.6` ft

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Summary of answers:
The first triangle has dimensions of 3 ft, 7.2 ft, and 7.8 ft.
The second triangle has dimensions of 9 ft, 21.6 ft, and 23.4 ft.