Question

1. the diagram shows a right - triangle and three squares. the area of the largest square is 55 units². which combination could be the areas of the smaller squares? there could be more than one answer. (a) 12 & 43 (b) 14 & 40 (c) 16 & 37 b) for each answer you selected above, how long would the side lengths of the triangle be?

Explanation:

Step1: Recall Pythagorean theorem

In a right - triangle, if the sides of the squares adjacent to the right - triangle have areas \(a\) and \(b\), and the area of the square on the hypotenuse is \(c\), then \(a + b=c\). We know \(c = 55\), and we need to check which pairs satisfy \(a + b=55\).

Step2: Check option (a)

For \(a = 12\) and \(b = 43\), \(12+43 = 55\).

Step3: Check option (b)

For \(a = 14\) and \(b = 40\), \(14 + 40=54
eq55\).

Step4: Check option (c)

For \(a = 16\) and \(b = 37\), \(16+37 = 53
eq55\).

Step5: Find side - lengths if option (a) is correct

If the area of a square \(A = s^{2}\), where \(s\) is the side - length. For the square with area \(A_1 = 12\), \(s_1=\sqrt{12}=2\sqrt{3}\). For the square with area \(A_2 = 43\), \(s_2=\sqrt{43}\).

Answer:

a) 12 & 43; The side - lengths of the squares with areas 12 and 43 are \(\sqrt{12}=2\sqrt{3}\) and \(\sqrt{43}\) respectively.