Question

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a triangular piece of rubber is stretched equally from all sides, without distorting its shape, such that each side of the enlarged triangle is twice the length of the original side
the area of the triangle to times the original area
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Explanation:

Step1: Recall the formula for the area of a triangle

The area of a triangle is given by \(A = \frac{1}{2}bh\), where \(b\) is the base and \(h\) is the height.

Step2: Analyze the effect of side - length scaling on base and height

If each side of the triangle is scaled by a factor of \(k = 2\), then the base \(b\) becomes \(b'=2b\) and the height \(h\) becomes \(h' = 2h\) (since the triangle is stretched without distorting its shape, the ratio of base to height remains the same).

Step3: Calculate the new area

The new area \(A'=\frac{1}{2}(2b)(2h)\). Using the associative property of multiplication, \(A'=\frac{1}{2}\times4bh = 4\times(\frac{1}{2}bh)\). Since \(A=\frac{1}{2}bh\), we have \(A' = 4A\).

Answer:

The area of the triangle increases to \(4\) times the original area.