Question

"The sides of a triangle have lengths of 4 mm, 6 mm, and 8 mm. In a similar triangle, the shortest side has the length (x - 2) mm, and the longest side has the length (x + 3) mm. What is the value of x? (You can draw a picture to aid in your solution.)"

Answer 1

The value of x is determined by setting up a proportion based on the sides of the similar triangles, leading to the equation (2(x - 2) = x + 3). Upon solving this equation, we find that x equals 7.

Step-by-step explanation:

In this mathematics problem, we are dealing with similar triangles. Because the triangles are similar, the sides of each triangle are proportional, and we can set up a proportion to find the value of x. In the original triangle, the sides are 4 mm, 6 mm, and 8 mm. Looking at the smallest and largest sides, we see that the ratio of the longest to the shortest side is 8/4 = 2. The proportion for the larger, similar triangle will be (x + 3) / (x - 2), and we want this to also equal 2, following the pattern of the original triangle.

Setting up the proportion, we get 2 = (x + 3) / (x - 2). To solve for x, we cross-multiply:
2(x - 2) = x + 3
2x - 4 = x + 3

Solving for x, subtract x from both sides and add 4 to both sides:
x - x - 4 + 4 = x - x + 3 + 4

x = 7.