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Brandon said that if $a$ is a positive real number, then $3 a, 4 a,$ and 5$a$ are the lengths of thesides of a right triangle. Therefore, $3 \sqrt{2}, 4 \sqrt{2},$ and 5$\sqrt{2}$ are the lengths of the sides of a right triangle. Do you agree with Brandon? Justify your answer.

Yes

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Chapter 3

REAL NUMBERS AND RADICALS

Section 5

Multiplying Radicals

Whole which of Numbers

Fractions and Mixed Numbers

Decimals

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So our question here is that if we know that a 345 triangle is in fact a right triangle is three bird to forward to fiber to also a right triangle if we just take all these sides and multiply by route to and there's a lot of different ways that we could answer this question. But the way I'm to do it is by comparing the ratios of the sides. So I'm going to label each side as a B or C just so we can map them together. Thes two are the same side. For example, the two that have the letter a um and so I'm gonna compare for the original right triangle will do it in green. A over B is equal to 3/4, and in our new triangle, the one that we think might be a right triangle. A over B is equal to three routes, two divided by four route to. But if we simplify, we realized that the two were choose just ah, they cancel out, leaving us with 3/4. So these are actually the same fracture, the same ratio. We can check the other sides to we could do be oversee that would be equal to 4/5 in our original triangle and in our new triangle, be oversee is equal to four Route two divided by five route to which the route twos cancel out, leaving us with four fifth. So that also shows us that those ratios air the same and we'll just do the last one. For good measure, we're gonna do the ratio of a oversee, which in our original triangle is 3/5. In our new triangle, a oversee is equal to three route to divided by five route to which you might be able to guess the root who's cancel out to leave us with just three fifths. So using our method of ratios, we get the answer that, yes, this triangle is a right triangle.

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