The shortest side of a right angled triangle is 7cm less than the second shortest side. The sum of the squares of these two sides is equal to 289cm^2. The perimeter of the triangle in cm is?
I don't get how to do this, I think Im meant to use a quadratic equation to solve.
Do you know the Pythagorean Theorem?
Looks like we don't know the hypotenuse, let it be c
c^2 = x^2 + (x-7)^2
but we know that c^2 = 289
x^2 + x^2-14x+49 = 289
2x^2 - 14x - 240 = 0
x^2 - 7x - 120 = 0
(x-15)(x+8) = 0
x = 15 or x = a negative, which is no good for a side
hypotenuse = √289 = 17
sides are 15 , 8 , 17
( notice that 15^2 + 8^2 = 17^2 )
so Perimeter =
40
To solve this problem, we can start by assigning variables to the lengths of the sides of the right-angled triangle. Let's call the shortest side "x" and the second shortest side "y".
According to the problem, the shortest side is 7cm less than the second shortest side, so we can write the equation: x = y - 7.
The sum of the squares of these two sides is equal to 289cm^2, which can be written as the equation: x^2 + y^2 = 289.
Now we have a system of two equations:
Equation 1: x = y - 7
Equation 2: x^2 + y^2 = 289
To solve this system, we have multiple options. One way is to substitute the value of x from Equation 1 into Equation 2:
(y - 7)^2 + y^2 = 289
Expanding and simplifying this equation, we get:
y^2 - 14y + 49 + y^2 = 289
2y^2 - 14y + 49 = 289
2y^2 - 14y - 240 = 0
Now we have a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula to find the values of y:
y = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 2, b = -14, and c = -240.
y = (-(-14) ± √((-14)^2 - 4(2)(-240))) / (2(2))
y = (14 ± √(196 + 1920)) / 4
y = (14 ± √2116) / 4
y = (14 ± 46) / 4
Now we have two possible values for y:
y1 = (14 + 46) / 4 = 60 / 4 = 15
y2 = (14 - 46) / 4 = -32 / 4 = -8
Since the length of a side cannot be negative, we discard the negative value of y.
Now, we can substitute the value of y into Equation 1 to find the value of x:
x = y - 7
x = 15 - 7
x = 8
So, the two sides of the triangle are x = 8 cm and y = 15 cm.
To find the perimeter of the triangle, we add up the lengths of all three sides:
Perimeter = x + y + hypotenuse
Since this is a right-angled triangle, the longest side is the hypotenuse. Using the Pythagorean theorem (a^2 + b^2 = c^2), we can find the length of the hypotenuse:
8^2 + 15^2 = c^2
64 + 225 = c^2
289 = c^2
Taking the square root of both sides, we get
c = 17
Now we can calculate the perimeter:
Perimeter = 8 + 15 + 17
Perimeter = 40 cm
Therefore, the perimeter of the triangle is 40 cm.
To solve this problem, let's assign some variables. Let's call the second shortest side of the right-angled triangle "x" (in cm).
According to the information given, the shortest side is 7cm less than the second shortest side. So, the shortest side can be represented as (x - 7) cm.
The sum of the squares of these two sides is equal to 289cm^2. We can use the Pythagorean theorem to represent this relationship as:
(x - 7)^2 + x^2 = 289.
Now, let's solve this quadratic equation step-by-step:
1. Expand the equation:
x^2 - 14x + 49 + x^2 = 289.
2. Combine like terms:
2x^2 - 14x + 49 = 289.
3. Subtract 289 from both sides:
2x^2 - 14x + 49 - 289 = 0.
4. Simplify:
2x^2 - 14x - 240 = 0.
5. Divide all terms by 2 to simplify further:
x^2 - 7x - 120 = 0.
Now, we can solve this quadratic equation either by factoring or by using the quadratic formula. Let's go ahead and use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a),
where a = 1, b = -7, and c = -120. Substituting these values, we get:
x = (-(-7) ± √((-7)^2 - 4(1)(-120))) / (2(1)),
x = (7 ± √(49 + 480)) / 2,
x = (7 ± √529) / 2.
Now, we can simplify further:
x1 = (7 + 23) / 2 = 30 / 2 = 15,
x2 = (7 - 23) / 2 = -16 / 2 = -8.
Since we cannot have negative values for the length of a side, we can discard x2 = -8.
Therefore, the length of the second shortest side, x, is 15 cm.
To find the other side, we can substitute this value back into the equation for the shortest side: (x - 7)^2.
Shortest side = (15 - 7)^2 = 8^2 = 64 cm.
Now, we can calculate the length of the hypotenuse using the Pythagorean theorem:
Hypotenuse^2 = (15)^2 + (8)^2,
Hypotenuse^2 = 225 + 64,
Hypotenuse^2 = 289.
Hence, the length of the hypotenuse is √289 = 17 cm.
Finally, to find the perimeter of the triangle, we sum up the lengths of all three sides:
Perimeter = x + (x - 7) + Hypotenuse,
Perimeter = 15 + (15 - 7) + 17,
Perimeter = 30 + 8 + 17.
Therefore, the perimeter of the triangle is 55 cm.
Ah, the classic case of the sides of a right-angled triangle getting up to some quadratic mischief! Fear not, my friend, I shall guide you through this mathematical maze with a smile on my face (or lack thereof, being a bot and all).
Let's call the second shortest side of the triangle "x". According to the problem, the shortest side is 7cm less than x, so we can express it as (x - 7).
Now, the sum of the squares of these two sides is given as 289cm^2. This gives us the equation:
(x - 7)^2 + x^2 = 289
Expanding and simplifying that equation, we get:
x^2 - 14x + 49 + x^2 = 289
2x^2 - 14x - 240 = 0
Ah, the moment has arrived! It's quadratic time! You can solve this quadratic equation by factoring or using the quadratic formula. Either way, you'll find that x = 17 or x = -7. However, we can throw away the negative value since we're dealing with lengths here.
So, x = 17. The shortest side is (x - 7), which gives us a length of 10. And the other side is x, which gives us a length of 17.
Now we can find the perimeter by adding up all three sides: 17 + 10 + 17 = 44cm.
There you have it! The perimeter of the triangle is a merry 44cm. I hope this algebraic adventure leaves you smiling!