The measures of the sides of a right triangle form an arithmetic sequence of common difference 2. What are these measures?
Linda, Anne,Denise each decided to knit a scarf. Linda knit one 1 yard long Anne knit one 4 feet Denise knit one 2 yards how many did they knit altogether
Well, in this case, let's call the shortest side of the right triangle "x". Since the side lengths form an arithmetic sequence with a common difference of 2, the other two sides would be x + 2 and x + 4. So, we have a right triangle with side lengths x, x + 2, and x + 4. Just remember, though, triangles aren't great at doing arithmetic – they're more into angles!
Let's denote the measures of the sides of the right triangle as a, a+2, and a+4, where a is the first term of the arithmetic sequence.
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. Applying this to our triangle:
(a+4)^2 = a^2 + (a+2)^2
Expanding this equation, we get:
a^2 + 8a + 16 = a^2 + a^2 + 4a + 4
Simplifying, we have:
8a + 16 = 2a^2 + 4a + 4
Rearranging the terms, we get a quadratic equation:
2a^2 + 4a + 4 - 8a - 16 = 0
Simplifying further:
2a^2 - 4a - 12 = 0
Dividing the equation by 2 gives:
a^2 - 2a - 6 = 0
Now, we can solve this quadratic equation using factoring or the quadratic formula.
Factoring is the preferred method here. We look for two numbers that multiply to give -6 and add up to -2 (the coefficient of 'a'). The numbers that fit these conditions are -3 and 1.
Therefore, the factored form of the equation is:
(a - 3)(a + 1) = 0
Setting each factor equal to zero and solving for a, we have:
a - 3 = 0 or a + 1 = 0
a = 3 or a = -1
Since we are dealing with the measures of sides, we disregard the negative value, leaving us with a = 3.
Thus, the measures of the sides of the right triangle are:
Side 1: a = 3 units
Side 2: a + 2 = 3 + 2 = 5 units
Side 3: a + 4 = 3 + 4 = 7 units
To find the measures of the sides of a right triangle with an arithmetic sequence, we can use the properties of right triangles.
Let's assume that the three sides of the right triangle are:
a - the shortest side (also known as the base),
b - the middle side, and
c - the longest side (also known as the hypotenuse).
Since the sides form an arithmetic sequence with a common difference of 2, we can express them as:
a, a + 2, a + 4
Now, let's apply the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + (a + 2)^2 = (a + 4)^2
Expanding and simplifying:
a^2 + a^2 + 4a + 4 = a^2 + 8a + 16
Combining like terms:
2a^2 + 4a + 4 = a^2 + 8a + 16
Moving all terms to one side:
2a^2 + 4a - a^2 - 8a - 12 = 0
Simplifying further:
a^2 - 4a - 12 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or using the quadratic formula. Factoring is the simplest method in this case:
(a - 6)(a + 2) = 0
Setting each factor equal to zero:
a - 6 = 0 or a + 2 = 0
Solving for a:
a = 6 or a = -2
Since the length of a side cannot be negative, we discard a = -2.
Therefore, the shortest side (base) of the right triangle is a = 6.
The other two sides can be found by adding 2 and 4, respectively:
a + 2 = 6 + 2 = 8
a + 4 = 6 + 4 = 10
So, the measures of the sides of the right triangle are:
6, 8, 10
(x-2)^2 + x^2 = (x+2)^2
2x^2-4x+4 = x^2+4x+4
x^2-8x=0
x = 0 or 8
0 is out, so the sides are
6,8,10