1. The sides of a triangle have lenghts x, x+4, and 20. Specify those values of x for which the triangle is acute with the longest side 20.
2. use the information to decide if triangle ABC is acute, right, or obtuse.
AC=13, BC= sq. rt. 34, CD=3
>>i know this is obtuse but why?
3. If x and y are positive numbers with x>y, show that a triangle with sides of lenghts 2xy, x^2 - y^2, and x^2 + y^2 is always a right triangle.
I HATE WORD PROBLEMS!! Thanks for helping =]
A right triangle with largest side 20 would have
x^2 + (x+4)^2 = 400
x^2 + 4x + 8 = 200
x^2 + 4x -192 = 0
(x-12)(x+16) = 0
The only root that makes sense (by being positive) is x = 12. That means that you have a right triangle if the sides are 12 and 16. (The x = 12 case). If x>12, the triangle is acute, meaning thet the largest angle is less than 90 degrees. If x<12, it is obtuse. You should be able to convince yourself of that by drawing the figure or by using the law of cosines.
It is not obtuse; it is impossible because one side is longer than the sum of the other two.
You also posted this problem separately. I gave a more complete answer there.
Let the first side be A, the second side b and the third side C.
Note that
(x^2 - y^2)^2 + (2xy)^2 = x^4 + 2x^2y^2 + y^2 = (x^2 + y^2)^2
Therefore the relationship A^2 + B^2 = C^2 is obeyed. This is only true for a right triangle.
i hate your guy's page bacause it dosnt come up and say the value of a skinny and a bit
No problem! I'll break down each question and explain how to solve it step by step.
1. To determine the values of x for which the triangle is acute with the longest side being 20, we need to apply the Triangle Inequality Theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In this case, we have the triangle with side lengths x, x+4, and 20. To ensure the triangle is acute, we need to compare the sum of the two smaller sides (x and x+4) with the longest side (20). The inequality is:
x + (x+4) > 20
Simplifying this inequality, we get:
2x + 4 > 20
Subtracting 4 from both sides gives:
2x > 16
Dividing both sides by 2, we obtain:
x > 8
Therefore, for the triangle to be acute with the longest side being 20, the value of x must be greater than 8.
2. To determine if triangle ABC is acute, right, or obtuse, we need to employ the Pythagorean Theorem. The Pythagorean Theorem states that for any right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, we have side lengths AC = 13, BC = √34, and CD = 3. To determine if the triangle is acute, right, or obtuse, we need to compare the squares of the lengths of AC and BC with the square of the length of the longest side (CD).
Calculating the squares of AC and BC respectively, we have:
AC^2 = 13^2 = 169
BC^2 = (√34)^2 = 34
Comparing these values with the square of CD (which is 3^2 = 9), we see that 169 + 34 = 203 is greater than 9. Therefore, the triangle ABC is obtuse because the sum of the squares of the two shorter sides is greater than the square of the longest side.
3. To show that a triangle with side lengths of 2xy, x^2 - y^2, and x^2 + y^2 is always a right triangle, we first need to determine the relationship between these side lengths.
We can apply the Pythagorean Theorem again. According to the theorem, for any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, the side lengths are 2xy, x^2 - y^2, and x^2 + y^2. We want to show that (2xy)^2 + (x^2 - y^2)^2 = (x^2 + y^2)^2.
Expanding and simplifying, we have:
4x^2y^2 + (x^4 - 2x^2y^2 + y^4) = x^4 + 2x^2y^2 + y^4
Cancelling out the common terms, we get:
4x^2y^2 - 2x^2y^2 = 2x^2y^2
Simplifying further, we have:
2x^2y^2 = 2x^2y^2
Since both sides of the equation are equal, we have shown that the given triangle always satisfies the Pythagorean Theorem. Therefore, the triangle with side lengths 2xy, x^2 - y^2, and x^2 + y^2 is always a right triangle.
I hope this explanation helps you understand how to solve these types of problems! Remember, practice makes perfect, so keep practicing and word problems will become easier over time.