Question

1. two out of three. if a right triangle has legs of length 1 and 2, what is the length of the hypotenuse? if it has one leg of length 1 and a hypotenuse of length 3, what is the length of the other leg? 3. hypotenuse hype. if a right triangle has legs of length 1 and x, what is the length of the hypotenuse? 4. assessing area. suppose you know the base of a rectangle has a length of 4 inches and a diagonal has a length of 5 inches. find the area of the rectangle. 5. squares all around. how does the figure below relate to the pythagorean theorem? 6. operating on the triangle. using a straightedge, draw a random triangle. now carefully cut it out. next amputate the angles by snipping through adjacent sides. now move the angles together so the vertices all touch and the edges meet. what do you conclude about the sum of the angles of a triangle? try this procedure with triangles having different dimensions. solidifying ideas

Explanation:

Step1: Recall Pythagorean theorem

For a right - triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\).

Step2: Solve question 2

Given \(a = 1\) and \(b = 2\), then \(c=\sqrt{a^{2}+b^{2}}=\sqrt{1^{2}+2^{2}}=\sqrt{1 + 4}=\sqrt{5}\).

Step3: Solve question 3

Given \(a = 1\) and \(c = 3\), then \(b=\sqrt{c^{2}-a^{2}}=\sqrt{3^{2}-1^{2}}=\sqrt{9 - 1}=\sqrt{8}=2\sqrt{2}\).

Step4: Solve question 4

In a rectangle with length \(l = 4\) and diagonal \(d = 5\), the width \(w=\sqrt{d^{2}-l^{2}}=\sqrt{5^{2}-4^{2}}=\sqrt{25 - 16}=\sqrt{9}=3\). The area of the rectangle \(A=l\times w=4\times3 = 12\) square inches.

Step5: Analyze figure related to Pythagorean theorem

The squares around the right - triangle satisfy the Pythagorean theorem. The sum of the areas of the two smaller squares (constructed on the legs of the right - triangle) is equal to the area of the larger square (constructed on the hypotenuse).

Step6: Find sum of angles of a triangle

When we move the angles of a triangle together so the vertices all touch and the edges meet, we find that the sum of the interior angles of a triangle is \(180^{\circ}\).

Answer:

1. The length of the hypotenuse is \(\sqrt{5}\).
2. The length of the other leg is \(2\sqrt{2}\).
3. The area of the rectangle is 12 square inches.
4. The sum of the areas of the two smaller squares around the right - triangle (constructed on the legs) is equal to the area of the larger square (constructed on the hypotenuse), which is related to the Pythagorean theorem.
5. The sum of the interior angles of a triangle is \(180^{\circ}\).