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2 years ago

Determine whether each of the following triangles is a right-angled triangle

Mathematics

1 answer:

Fantom [35]2 years ago

4 0

Answer:

Yes

Step-by-step explanation:

To determine if this is a right triangle, use the pythagorean theorem and see if a^2 + b^2 is equal to c^2.

a^2 + b^2 = c^2

12^2 + 16^2 = 20^2

Evaluate the exponents.

144 + 256 = 400

Add them together.

400 = 400

Since a^2 and b^2 add up to equal c^2, this is a right triangle.

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Set

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$y=\rho\sin\theta\sin\varphi$

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$\mathrm dV=\mathrm dx\,\mathrm dy\,\mathrm dz=\rho^2\sin\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\varphi$

The integral is then

$\displaystyle\iiint_Exyz\,\mathrm dV=\int_{\varphi=0}^{\varphi=\pi/3}\int_{\theta=0}^{\theta=2\pi}\int_{\rho=4}^{\rho=6}\rho^5\cos\theta\sin\theta\sin^3\varphi\cos\varphi\,\mathrm d\rho\,\mathrm d\theta\,\mathrm d\rho$

$=\displaystyle\left(\int_0^{\pi/3}\sin^3\varphi\cos\varphi\,\mathrm d\varphi\right)\underbrace{\left(\int_0^{2\pi}\cos\theta\sin\theta\,\mathrm d\theta\right)}_0\left(\int_4^6\rho^5\,\mathrm d\rho\right)$

and so evaluates to 0.

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