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aalyn [17]
2 years ago
7

suppose someone tells you that she has a triangle with sides having lengths 2.6,8.1 and 8.6. Is this a right triangle? why or wh

y not? Is there an angle in the triangle larger than 90 degree?
Mathematics
1 answer:
sveta [45]2 years ago
3 0
If you're only provided with the lengths of a triangle, and you're asked to determine whether or not the triangle is right or not, you'll need to rely on the Pythagorean Theorem to help you out. In case you're rusty on it, the Pythagorean Theorem defines a relationship between the <em>legs</em> of a right triangle and its <em>hypotenuse</em>, the side opposite its right angle. That relationship is a² + b² = c², where a and b are the legs of the triangle, and c is its hypotenuse. To see if our triangle fits that requirement, we'll have to substitute its lengths into the equation.

How do we determine which length is the hypotenuse, though? Knowledge that the hypotenuse is always the longest length of a right triangle helps here, as we can clearly observe that 8.6 is the longest we've been given for this problem. The order we pick the legs in doesn't matter, since addition is commutative, and we'll get the same result regardless of the order we're adding a and b.

So, substituting our values in, we have:

(2.6)² + (8.1)² = (8.6)²

Performing the necessary calculations, we have:

6.76 + 65.61 = 73.96
72.37 ≠ 73.96

Failing this, we know that our triangle cannot be right, but we <em>do </em>know that 72.37 < 73.96, which tells us something about what kind of triangle it is. Imagine taking a regular right triangle and stretching its hypotenuse, keeping the legs a and b the same length. This has the fact of <em>increasing the angle between a and b</em>. Since the angle was already 90°, and it's only increased since then, we know that the triangle has to be <em>obtuse</em>, which is to say: yes, there's an angle in it larger than 90°.
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The easiest way to solve this, is expressing the division as a fraction, and then use divisibility rules to simplify our fraction.

Remember that saying 6476 divided by 12 is the same as \frac{6476}{12}. Just like that we have our fraction. Now to simplify it, we are going to use divisibility rules

Remember that to simplify a fraction using divisibility rules, both numerator and numerator must by divisible by the same number.

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A number is divisible by 2 if and only if it ends in 0, 2, 4, 6, or 8. In other words, a number is divisible by 2 if its last digit is even or zero.

Notice that 6476 ends in 6, so it's divisible by 2, and 12 ends in 2, so it's also divisible by 2. Since both the numerator and denominator of our fraction are divisible by 2, we can simplify our fraction dividing both numerator and denominator by 2. To divide by 2, we just need to find the half of each number. The half of 6476 is 3238 and the half of 12 is 6, so we can simplify our fraction as follows:

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3238 ends in 8, so it's divisible by 2; 6 ends in 6, so it's divisible by 2. The half of 3238 is 1619 and the half of 6 is 3, so we can simplify our fraction even more:

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Now, 1619 does end in zero or an even number, so it's not divisible by 2; similarly, 3 is not even even, so it's not divisible by 2. Let's check if they are divisible by 3

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Let's check if the numerator of our fraction, 1619, is divisible by 3:

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Now, let's check if our numerator, 3, is divisible by 3:

3 is a multiple of 3, so its divisible by 3.

Since our numerator is divisible by 3 but our denominator is not, we can't further simplify our fraction.

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