### All Problems of the Week – And Solutions

Thank you for your comments and answers to the problems of the week!  We will be back in September with more posts and problems of the week. Enjoy the summer!

Using your imagination, show or describe 10 ways to connect dots A and B. 1)    What information is needed for you to calculate the area of the square?

2)    How can the two triangle measurements help you figure out the area of the square?

3)    Figure ADE is a right triangle. Where is the right angle? Where is the hypotenuse?

4)    How can you use the Pythagorean Theorem to figure out the length of line segment AD?

5)    What is the length of line segment AD? What is the length of line segment AB?

6)    What is the area of the square?

Figure ABCD is a square. Figure ADE is a right triangle. What is the area of the square? 1) What information is needed to calculate the perimeter of the triangle? What information do you already have?

2) How can you figure out the length of line segment AD? You are given the area of square ABCD; how can that help you discover the length of AD?

3) How is the area of a square or rectangle usually calculated? How can you calculate this area, if you don’t know the length of any of the sides?

4) But the area has already been calculated. Can you think backwards to figure out what the length of the sides must be?  Are the sides all the same length? How do you know for sure? What number times itself equals 4? What is the square root of 4 square meters? What is the length of AD?

5) How can you figure out the length of line segment AE? You know the length of AD and DE, which are two sides of the triangle. What kind of triangle is it? Which side of the right triangle is the hypotenuse? If this is a right triangle, can you use the Pythagorean Theorem to calculate the length of the hypotenuse? What is the Pythagorean Theorem: Is it a + b = c? Is it a² + b² = c²? Does the Pythagorean Theorem tell you the length of the hypotenuse, or the length of the hypotenuse squared?

6) Extra hint for students who already know about 3-4-5 right triangles: double the measurements of AD and DE; what do you notice?

Figure ABCD is a square with an area of 4 square meters. Figure ADE is a right triangle. One side measures 1.5 meters as indicated. What is the perimeter of the triangle?  What prior concepts or procedures do students need in order to solve this problem? 1) What information is needed for you to calculate the circumference of the circle? Is that information presented on the drawing? Can you figure out the needed information from the information that is given to you in the picture? How will you do it?

2) What does “isosceles” mean? What is the length of side ED of the triangle ADE? How can you use the Pythagorean Theorem (a2 + b2 = c2) to figure out the length of side AD? What is the length of it? Is Line AD part of the triangle? Is it also part of the square?

3)  Do we know the exact value of the square root of 2? What do you get if you multiply the square root of 2 times the square root of 2? How do we calculate the area of square ABCD? What is the area?

4) Draw a diagonal from A to C. How can you use the Pythagorean Theorem to figure out the length of Line AC? What is the length of it? Is Line AD part of a triangle? Is it also part of the circle? Is the diagonal Line AC the same as the diameter of the circle? How do you know? If you know the length of the diameter, how  can you calculate the circumference of the circle? What is it?

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