Question

A triangle has a perimeter of 45cm. One side is twice as long as the shortest side. The remaining side is 25cm less than the square of the shortest side. Find the length of all the sides of the triangle.

Answer 1

The lengths of the sides of the triangle are 7 cm, 14 cm, and 24 cm, calculated by setting up and solving a quadratic equation based on the given perimeter and the relationships among the sides.

Step-by-step explanation:

To solve the triangle problem, we need to set up equations based on the given information. Let the shortest side of the triangle be x cm. Thus, the second side is 2x cm, and the third side is x2 - 25 cm. The perimeter of the triangle is the sum of the lengths of all sides, which equals 45 cm.

Therefore, our equation based on the perimeter is:

x + 2x + (x2 - 25) = 45

Simplifying this equation:

x2 + 3x - 25 - 45 = 0

x2 + 3x - 70 = 0

Solving this quadratic equation, we find that x = 7 or x = -10. Since a side length cannot be negative, x = 7 cm is the length of the shortest side.

Finally, the lengths of the sides of the triangle are:

• Shortest side: x = 7 cm,
• Second side: 2x = 2(7) cm = 14 cm,
• Third side: x2 - 25 = 72 - 25 cm = 49 - 25 cm = 24 cm.

Answer 2

The length of the sides of the triangle are: 7 cm, 14 cm, 24 cm Let's call the length of the shortest side of the triangle "s". With that in mind, let's create some equations summarizing what we know. "One side is twice as long as the shortest side" So one of the sides is 2s "The remaining side is 25cm less than the square of the shortest side." That makes the 3rd side: s^2 - 25 So the 3 sides we have have a length of s 2s s^2 - 25 And the final piece of the puzzle is "triangle has a perimeter of 45cm", so our final equation becomes s + 2s + (s^2 - 25) = 45 Now solve for s s + 2s + (s^2 - 25) = 45 s + 2s + s^2 - 25 = 45 3s + s^2 - 25 = 45 s^2 + 3s - 25 = 45 s^2 + 3s - 70 = 0 We now have a regular quadratic equation. We could use the quadratic formula to find it's roots. But let's do it the old fashioned way. Since the 3rd term is negative, the factorization will be of the form: (s + x)(s - y) Also since the coefficient of the s^2 is 1, the first terms of both factors will be simply s. And since the 2nd term has a coefficient of 3, we need to find 2 factors of the 3rd term that have a difference of 3. The numbers 7 and 10 are quite suitable. So we have (s + 10)(s - 7) as the factorization, which means that s has a value of either -10, or 7. Since a negative length doesn't make sense for this problem, we'll use the positive value of 7 as the length of the shortest side. Now since we know the shortest side is 7. The side that's twice as long is 2*7 = 14. And the third side is 25 less than 7 squared, so 7^2 - 25 = 49 - 25 = 24. So our sides are 7, 14, 24 And finally, as a quick check, let's add them together to make sure the perimeter is correct 7 + 14 + 24 = 45 And it is correct.