Figure ABCD is a square. Figure ADE is a right triangle. What is the area of the square?

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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?

Figure ABCD is a square. Points A, B, C, and D are all located on the circumference of the circle. Figure ADE is an isosceles right triangle. One side measures 1 meter.

What is the area of the square? What is the area of the circle?  (Use 3.14 for ∏.)

What prior knowledge/skills do students need to bring to this problem?

1) List all the features of a circle: (All circles have a center, a radius, a diameter, a circumference, and an area; the radius is half of the diameter; A = πr2; C = πD)

2) List all the features of a rectangle: (All rectangles have 4 sides and 4 right angles, a perimeter, and an area; the top and bottom sides are parallel and equal in length; the right side and left side are parallel and of equal length; L•W = A; P = 2L + 2W)

3) How can you figure out the length of Line DC?

4) How can you create a right angle in this drawing?

5) If you draw a line connecting points A and C, how would you figure out its length? (Hint: Pythagoras…)

6) Would Line AC pass through the center of the circle? How do you know?

7) Is Line AC the same as the diameter of the circle?

8 ) What information do you need to figure out the circumference of the circle? Do you have that information? Can you figure it out?

9) When calculating the circumference, use 22/7 for π, and use cross-canceling so that you can work with smaller numbers,

10) What is the length of the circumference of the circle?

Building on the last two problems, what do students need to notice in the diagram, what shapes, properties, and relationships do they need to consider to solve this problem? Then solve it.

The area of rectangle ABCD is 588 cm². The left side is 21 cm. What is the circumference of the circle? Use Π = 22/7.