Question

dearies
e drop-down menus to explain your reasoning.
triangle formed by three squares with areas of 9 square units, 16 square units, and choose... square units is a(n) choose... triangle with side lengths of
choose...

Explanation:

Step1: Recall the Pythagorean theorem for squares on triangle sides

For a right - triangle, if the areas of the squares on the two legs are \(A_1\) and \(A_2\), and the area of the square on the hypotenuse is \(A_3\), then \(A_3=A_1 + A_2\) (by the Pythagorean theorem, since the area of a square with side length \(s\) is \(s^2\), and if the side lengths of the triangle are \(a\), \(b\), \(c\) (where \(c\) is the hypotenuse), then \(a^{2}=A_1\), \(b^{2}=A_2\), \(c^{2}=A_3\) and \(a^{2}+b^{2}=c^{2}\)).

Given \(A_1 = 9\) square units and \(A_2=16\) square units. Then \(A_3=9 + 16=25\) square units.

Step2: Determine the type of triangle

Since \(9 + 16=25\), the triangle satisfies the Pythagorean theorem (\(a^{2}+b^{2}=c^{2}\) where \(a^{2}=9\), \(b^{2}=16\), \(c^{2}=25\)), so it is a right - triangle.

Step3: Find the side lengths

If the area of a square is \(s^{2}\), then for the square with area 9, \(s=\sqrt{9} = 3\) units; for the square with area 16, \(s=\sqrt{16}=4\) units; for the square with area 25, \(s=\sqrt{25}=5\) units. So the side lengths of the triangle are 3 units, 4 units, and 5 units.

Answer:

The area of the third square is 25 square units, the triangle is a right triangle with side lengths of 3 units, 4 units, and 5 units. So the first "Choose" should be 25, the second "Choose" should be "right", and the side lengths are 3, 4, 5 (in appropriate drop - down selections).