SECTION 3.5 TRANSLATING SENTENCES IN Geometry An isosceles triangle has two sides of equal length. The length of one of the equal sides is 2m more than three times the length of the third side. If the perimeter is 46m, find the length of each side.
The Correct Answer and Explanation is:
Correct Answer:
Let the length of the third side be x meters.
Since the triangle is isosceles, the other two sides (which are equal) are each 3x + 2 meters.
The perimeter of a triangle is the sum of all its sides, so:x+(3x+2)+(3x+2)=46x + (3x + 2) + (3x + 2) = 46x+(3x+2)+(3x+2)=46
Simplify the equation:x+3x+2+3x+2=46⇒7x+4=46⇒7x=42⇒x=6x + 3x + 2 + 3x + 2 = 46 \Rightarrow 7x + 4 = 46 \Rightarrow 7x = 42 \Rightarrow x = 6x+3x+2+3x+2=46⇒7x+4=46⇒7x=42⇒x=6
Now substitute x = 6 back into the expressions for the sides:
- Third side: x = 6 meters
- Each of the equal sides: 3(6) + 2 = 20 meters
So, the lengths of the sides are 6 meters, 20 meters, and 20 meters.
Explanation
To solve this problem, we begin by understanding the properties of an isosceles triangle. In geometry, an isosceles triangle is defined as a triangle that has at least two sides of equal length. The problem tells us that the length of one of the equal sides is 2 meters more than three times the length of the third side. Additionally, we are given that the perimeter of the triangle is 46 meters. Our goal is to find the length of each side.
We start by assigning a variable to the unknown quantity. Let the third side (the one that is different in length) be x meters. The two equal sides are described as 2 meters more than three times this third side. So, each of the equal sides can be written as 3x + 2 meters.
Next, we set up an equation for the perimeter of the triangle. The perimeter is the sum of the lengths of all three sides, so we add x + (3x + 2) + (3x + 2) and set it equal to 46. This gives us the equation 7x + 4 = 46.
Solving this equation involves isolating the variable x. Subtracting 4 from both sides, we get 7x = 42, and dividing by 7, we find x = 6. This means the third side is 6 meters long. Substituting this value back into the expressions for the other sides, we find each equal side is 3(6) + 2 = 20 meters long.
Therefore, the triangle has side lengths of 6 meters, 20 meters, and 20 meters.
