: the perimeter of triangle abc is 26m bc is two -thirds as long as ab and it is also 2m longer than ac .find the lengths of the three sides
p = ab+bc+ac = 26
bc = 2/3 ab
bc = ac+2
ab + 2/3 ab + (2/3 ab - 2) = 26
ab (1 + 2/3 + 2/3) - 2 = 26
7/3 ab = 28
ab = 12
bc = 8
ac = 6
Well, to find the lengths of the sides, we need to use our mathematician skills... and maybe a pinch of clown logic!
Let's say that the length of AB is x meters. Since BC is two-thirds as long as AB, we can say that BC = (2/3)x meters.
Now, the problem tells us that BC is also 2 meters longer than AC. So we can write the equation: (2/3)x = AC + 2.
And finally, the perimeter of any triangle is the sum of its side lengths. In this case, it's 26 meters, so we can write the equation: x + (2/3)x + AC = 26.
Now, let's juggle these equations around a bit to solve for the lengths!
From the perimeter equation, we have: x + (2/3)x + AC = 26.
From the second equation, we have: (2/3)x = AC + 2.
Let's multiply the second equation by 3 to get rid of the fraction: 2x = 3(AC + 2).
Simplifying, we have: 2x = 3AC + 6.
Now, let's substitute this value of 2x into the first equation: (3AC + 6) + (2/3)(3AC + 6) + AC = 26.
Expanding and simplifying, we get: 3AC + 6 + 2AC + 4 + AC = 26.
Combining like terms, we have: 6AC + 10 = 26.
Subtracting 10 from both sides, we get: 6AC = 16.
Dividing both sides by 6, we find: AC = 16/6 = 8/3.
Finally, substituting this value of AC back into the second equation, we have: (2/3)x = 8/3 + 2.
Simplifying, we get: (2/3)x = 14/3.
Next, we multiply both sides by 3/2 to solve for x: x = (3/2)(14/3) = 7.
So, AB = x = 7 meters.
And BC = (2/3)x = (2/3)(7) = 14/3 meters.
Therefore, the three sides of the triangle have lengths: AB = 7 meters, BC = 14/3 meters, and AC = 8/3 meters.
Now, if you need any clown-approved dance moves to solve this math problem, just let me know! 🤡
Let's assign variables to the lengths of the three sides of triangle ABC. Let AB be x meters, BC be y meters, and AC be z meters.
According to the given information:
1. "The perimeter of triangle ABC is 26m." This means that the sum of the three sides must equal 26m.
x + y + z = 26 --(equation 1)
2. "BC is two-thirds as long as AB." This means that the length of BC is (2/3) times the length of AB.
y = (2/3)x --(equation 2)
3. "BC is also 2m longer than AC." This means that the length of BC is 2m more than the length of AC.
y = z + 2 --(equation 3)
We now have a system of three equations with three variables. We can solve this system of equations to find the values of x, y, and z.
Using equations (2) and (3), we can substitute the value of y from equation (2) into equation (3):
(2/3)x = z + 2
Using equations (1) and (2), we can substitute the value of y from equation (2) into equation (1):
x + (2/3)x + z = 26
Simplifying equation (1):
(5/3)x + z = 26 --(equation 4)
Now, we have two equations with two variables: equation (4) and equation (5).
To solve this system of equations, we can use substitution or elimination method:
Method 1: Substitution:
- Rearrange equation (2) to isolate z: z = (2/3)x - 2
- Substitute this value of z in equation (4): (5/3)x + ((2/3)x - 2) = 26
- Simplify and solve for x: (7/3)x = 28
- Divide both sides by (7/3): x = 12
- Substitute the value of x into equation (2) to find the value of y: y = (2/3)(12) = 8
- Substitute the value of x into equation (3) to find the value of z: z = 12 + 2 = 14
Method 2: Elimination
- Multiply equation (2) by 3 to get rid of fractions: 3y = 2x
- Multiply equation (3) by 3 to get rid of fractions: 3y = 3z + 6
- Now, we have two equations: 3y = 2x and 3y = 3z + 6
- Equate the two equations: 2x = 3z + 6
- Rearrange this equation: 2x - 3z = 6
- Substitute the value of y from equation (2) into equation (1): x + (2/3)x + z = 26
- Simplify: (5/3)x + z = 26
- Rearrange this equation: z = 26 - (5/3)x
- Substitute this value of z in the rearranged equation from earlier: 2x - 3(26 - (5/3)x) = 6
- Simplify and solve for x: x = 12
- Substitute the value of x into equation (2) to find the value of y: y = (2/3)(12) = 8
- Substitute the value of x into equation (3) to find the value of z: z = 12 + 2 = 14
Therefore, the lengths of the sides of triangle ABC are: AB = 12m, BC = 8m, and AC = 14m.
To find the lengths of the three sides of the triangle, let's assign variables to the lengths of the sides.
Let AB represent the length of side ab, BC represent the length of side bc, and AC represent the length of side ac.
We are given two pieces of information. First, we know that BC is two-thirds as long as AB. This can be written as: BC = (2/3)AB.
Second, we know that BC is also 2m longer than AC. This can be written as: BC = AC + 2.
Now, let's express the perimeter of the triangle in terms of AB, BC, and AC: perimeter = AB + BC + AC.
Since we know that the perimeter is 26m, we can write the equation as: 26 = AB + BC + AC.
Now, let's substitute the values we found earlier for BC and AC into the equation:
26 = AB + (2/3)AB + (AC + 2).
Simplifying this equation, we get:
26 = AB + (2/3)AB + AC + 2.
Combining like terms, we have:
26 = (5/3)AB + AC + 2.
To solve for the lengths of the sides, we need another equation involving AB and AC. Since we don't have any more given information, we cannot find the lengths of the sides without additional information.