Right Triangle Checks: Do These Sides Measure Up?

Hey guys! Ella here, and I'm on a mission to make sure our roof frames are built with perfect right triangles. Why are right triangles so important for roofs, you ask? Well, they provide the strongest and most stable structure, ensuring that your roof can withstand all sorts of weather and last for ages. Plus, they make all the angles line up just right, which is super important for, you know, keeping the rain out and the snow from piling up too much. Today, we're diving deep into the world of Pythagorean triples to figure out which sets of measurements actually form these essential geometric shapes. We'll be using that awesome theorem, the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Mathematically, it's expressed as $a^2 + b^2 = c^2$, where 'c' is always the hypotenuse. So, grab your calculators, maybe a trusty notebook, and let's crunch some numbers together to see if Ella's roof frames are rock solid!

The Pythagorean Theorem: Our Secret Weapon

Alright, let's get down to business with the Pythagorean theorem. This is our main tool for determining if a triangle is a right triangle. Remember, the theorem, $a^2 + b^2 = c^2$, only works for right triangles. So, if we're given three side lengths, we need to check if they satisfy this equation. The trickiest part, especially when you're dealing with potential right triangles, is identifying the hypotenuse. The hypotenuse is always the longest side of the triangle. So, before we start squaring things, we need to identify which of the given measurements is the longest one. Once we've found that, we'll designate it as 'c'. The other two sides will be our 'a' and 'b'. Then, we square 'a', square 'b', add them together, and see if that sum equals the square of 'c'. If it does, bam! We've got ourselves a right triangle. If it doesn't, then unfortunately, that set of measurements won't form a right triangle, and we'll have to go back to the drawing board, or in Ella's case, maybe get some different lumber. It's a straightforward process, but paying close attention to identifying the hypotenuse and doing the calculations correctly is key. Let's break down each option Ella is considering.

Option A: 10, 24, 26

First up, we have the measurements 10, 24, and 26. Our first step is to identify the longest side, which is clearly 26. So, we'll set $c = 26$. The other two sides are $a = 10$ and $b = 24$. Now, let's apply the Pythagorean theorem: $a^2 + b^2 = c^2$. We need to calculate $10^2 + 24^2$ and see if it equals $26^2$. Let's do the math:

$10^2 = 10 * 10 = 100$ $24^2 = 24 * 24 = 576$

Now, we add these together: $100 + 576 = 676$.

Next, let's square the hypotenuse: $26^2 = 26 * 26 = 676$.

Since $676 = 676$, the equation $a^2 + b^2 = c^2$ holds true! Awesome! This means that the measurements 10, 24, and 26 do indeed form a right triangle. So, Ella can definitely use this set of measurements for her roof frames. It's always great when things line up perfectly, right?

Option B: 8, $\sqrt{20}$, 25

Moving on to option B, we have the measurements 8, $\sqrt{20}$, and 25. First, let's identify the longest side. We know that 25 is a whole number. For $\sqrt{20}$, we know that $\sqrt{16} = 4$ and $\sqrt{25} = 5$, so $\sqrt{20}$ is somewhere between 4 and 5. Therefore, 25 is clearly the longest side. So, we set $c = 25$. Our other two sides are $a = 8$ and $b = \sqrt{20}$. Let's plug these into the Pythagorean theorem: $a^2 + b^2 = c^2$. We need to calculate $8^2 + (\sqrt{20})^2$ and see if it equals $25^2$. Let's get calculating:

$8^2 = 8 * 8 = 64$ $(\sqrt{20})^2 = 20$ (The square root and the square cancel each other out, which is super handy!)

Now, we add these together: $64 + 20 = 84$.

Next, let's square the hypotenuse: $25^2 = 25 * 25 = 625$.

Here, we see that $84 \neq 625$. The equation $a^2 + b^2 = c^2$ does not hold true. This means that the measurements 8, $\sqrt{20}$, and 25 do not form a right triangle. So, Ella should avoid using this set for her critical roof structures. It's a bummer when they don't work out, but knowing is half the battle, right?

Option C: 8, $\sqrt{161}$, 15

Next up is option C, with measurements 8, $\sqrt{161}$, and 15. Let's find the longest side. We have 8 and 15, which are straightforward. For $\sqrt{161}$, we know that $\sqrt{144} = 12$ and $\sqrt{169} = 13$. So, $\sqrt{161}$ is between 12 and 13. Comparing 8, $\sqrt{161}$ (approx. 12.something), and 15, it's clear that 15 is the longest side. So, $c = 15$. Our other two sides are $a = 8$ and $b = \sqrt{161}$. Now, we apply the Pythagorean theorem: $a^2 + b^2 = c^2$. We need to calculate $8^2 + (\sqrt{161})^2$ and compare it to $15^2$.

$8^2 = 8 * 8 = 64$ $(\sqrt{161})^2 = 161$

Adding them up: $64 + 161 = 225$.

Now, let's square the hypotenuse: $15^2 = 15 * 15 = 225$.

Eureka! Since $225 = 225$, the equation $a^2 + b^2 = c^2$ is satisfied. This tells us that the measurements 8, $\sqrt{161}$, and 15 do form a right triangle. Excellent news for Ella and her construction plans!

Option D: 20, 21, 29

Let's check out option D: 20, 21, and 29. We need to find the longest side. Comparing 20, 21, and 29, it's obvious that 29 is the longest side. So, we set $c = 29$. The other two sides are $a = 20$ and $b = 21$. Time to plug them into the Pythagorean theorem: $a^2 + b^2 = c^2$. We'll calculate $20^2 + 21^2$ and see if it equals $29^2$.

$20^2 = 20 * 20 = 400$ $21^2 = 21 * 21 = 441$

Adding these two results: $400 + 441 = 841$.

Now, let's find the square of the hypotenuse: $29^2 = 29 * 29 = 841$.

And there we have it! Since $841 = 841$, the equation $a^2 + b^2 = c^2$ is confirmed. This means that the measurements 20, 21, and 29 definitely form a right triangle. Another win for Ella's roof project!

Option E: 39, 80, 89

Finally, we've reached option E: 39, 80, and 89. Our first step, as always, is to identify the longest side. Comparing 39, 80, and 89, the longest side is 89. So, $c = 89$. The remaining two sides are $a = 39$ and $b = 80$. Let's apply the Pythagorean theorem: $a^2 + b^2 = c^2$. We need to calculate $39^2 + 80^2$ and check if it equals $89^2$.

$39^2 = 39 * 39 = 1521$ $80^2 = 80 * 80 = 6400$

Adding these values: $1521 + 6400 = 7921$.

Now, let's square the hypotenuse: $89^2 = 89 * 89 = 7921$.

Incredible! Since $7921 = 7921$, the Pythagorean theorem holds true for these measurements. This confirms that 39, 80, and 89 form a right triangle. What a great result for Ella!

The Verdict: Which Frames Are Right?

So, after all that number crunching, let's tally up the results. Ella is looking for measurements that form right triangles to ensure the stability and structural integrity of her roof frames. We tested each option using the Pythagorean theorem, $a^2 + b^2 = c^2$, where 'c' is the longest side.

• Option A (10, 24, 26): $10^2 + 24^2 = 100 + 576 = 676$, and $26^2 = 676$. Yes, this forms a right triangle.
• Option B (8, $\sqrt{20}$, 25): $8^2 + (\sqrt{20})^2 = 64 + 20 = 84$, and $25^2 = 625$. $84 \neq 625$. No, this does not form a right triangle.
• Option C (8, $\sqrt{161}$, 15): $8^2 + (\sqrt{161})^2 = 64 + 161 = 225$, and $15^2 = 225$. Yes, this forms a right triangle.
• Option D (20, 21, 29): $20^2 + 21^2 = 400 + 441 = 841$, and $29^2 = 841$. Yes, this forms a right triangle.
• Option E (39, 80, 89): $39^2 + 80^2 = 1521 + 6400 = 7921$, and $89^2 = 7921$. Yes, this forms a right triangle.

Therefore, the measures that form right triangles are A, C, D, and E. Ella can confidently use these sets of measurements for her roof framing projects, knowing she's building with stability and precision. Keep those calculations sharp, guys!