10.4.2 What Can You See?

Activity Symbol Activity: What Can You See?

Find the tangram puzzle you made in Section 10.3 and label it as shown in the image below.

(a) Which sets of lines appear to be parallel?

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Answer

There are four sets of parallel lines. AB is parallel to DC. This can be written as cap a times cap b parallel to cap d times cap c. Similarly, cap d times cap b parallel to cap h times cap j comma and cap a times cap i parallel to cap e times cap h. Also, AD, GI, and BC are all parallel to each other.

(b) Which lines are perpendicular?

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AB is perpendicular to AD and to BC. This can also be written as cap a times cap b up tack cap a times cap d comma and cap a times cap b up tack cap b times cap c. Similarly, cap d times cap c up tack cap a times cap d comma and cap d times cap c up tack cap b times cap c. Also, cap e times cap h up tack cap d times cap b comma cap e times cap h up tack cap h times cap j comma cap a times cap i up tack cap d times cap b, and cap a times cap i up tack cap h times cap j.

(c) How many triangles can you see in the puzzle?

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Answer

There are the seven triangles in the puzzle: prefix white up pointing triangle of cap a times cap b times cap d , prefix white up pointing triangle of cap d times cap b times cap c , prefix white up pointing triangle of cap a times cap b times cap f , prefix white up pointing triangle of cap a times cap d times cap f , prefix white up pointing triangle of cap d times cap e times cap h , prefix white up pointing triangle of cap f times cap g times cap i , and prefix white up pointing triangle of cap h times cap j times cap c. Give yourself a pat on the back if you found them all!

If you did not see all of these, trace the triangles by following the letters.

(d) What features are the same and what features are different in these triangles?

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Answer

All these triangles are right-angled. In each triangle, there is one right angle of 90º and then two other angles that are both half of 90º, which is 45º. If two figures have the same shape, but not necessarily the same size, the two figures are said to be similar. Figures are said to be similar if one is a scaled enlargement of the other. [ Note how the mathematical meaning of the word “similar” is different from and more precise than the everyday meaning. ]

For example, prefix white up pointing triangle of cap a times cap f times cap d is similar to prefix white up pointing triangle of cap f times cap g times cap i because the sides of prefix white up pointing triangle of cap a times cap f times cap d are all double the corresponding sides of prefix white up pointing triangle of cap f times cap g times cap i. Here, all the triangles in the puzzle are similar to each other. In general, two figures are similar if corresponding sides are in the same ratio and corresponding angles are equal. So the trapezoid FBJI is similar to the trapezoid JCDF.

In each triangle, two of the angles are the same, so the triangles are isosceles. You may also have spotted this by noting that each triangle has two sides of the same length. [ Shapes that are the exact same size and shape are said to be congruent. Can you spot any other congruent triangles in the puzzle? ]

prefix white up pointing triangle of cap d times cap e times cap h and prefix white up pointing triangle of cap f times cap g times cap i are the exact same size and shape. Such figures are said to be congruent—meaning that one would lie exactly on top of the other.

(e) What other shapes can you see?

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Answer

There are many other shapes in the diagram. Did you spot the parallelogram, GBJI, the squares ABCD and EFIH, and the trapezoids DHIF, EHIG, DBJH, BJHE, and DGIH?

Now, before you cut up the puzzle, label the pieces as shown below.

Now, cut along the thick lines on the puzzle to make seven pieces. Keep the pieces in an envelope or plastic container when you are not using them.

Activity Symbol Activity: Making Shapes

Find the square and the two smallest triangles (shapes 4, 5, and 6). Complete parts (a), (b), and (c) using these three shapes only.

(a) Make a rectangle.

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Answer

(a) Put the longest sides of the two triangles next to each other to make a square. Then put the squares next to each other to make a rectangle.

(b) Make a triangle.

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(b) Put the short side of triangle 4 along the top edge of the square and the short edge of triangle 6 along the side edge of the square to make a triangle.

(c) Make a parallelogram.

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(c) Take the rectangle you made in part (a). Slide the right hand triangle back over the square so that the short edge of the triangle is next to the far side of the square. This should make a parallelogram.

[ Can you make a square using five, six, and seven pieces too? ] Make a square using the specified number of pieces.

(d) Use two pieces.

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Answer

(d) Put the two long sides of the triangle together to make a square.

(e) Use three pieces.

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Answer

(e) Put two of the short sides of the two triangles together to form a triangle. Put the longest side of this triangle next to the longest side of the large triangle.

(f) Use four pieces.

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Answer

(f) Put the longest side of one of the small triangles against the longest side of the parallelogram. Put one of the shortest sides of the second small triangle against the shorter side of the parallelogram to form a triangle. Put the longest side of this triangle against the longest side of the large triangle.

Optional Brain Stretcher

Using all seven tangram pieces for each shape, can you make the following shapes? What other shapes or pictures can you make?

This section has introduced a lot of mathematical vocabulary that you will need later in this unit. After you take notes on these terms, it is worth looking back over these definitions frequently—perhaps at the start of each study session so that they become familiar. Try to practise using the terms elsewhere in your everyday life: Can you spot any isosceles triangles, parallel lines, or similar shapes anywhere? This may be something you can get your family or friends to help with as well.

10.4.1 Everyday Shapes

10.5 Perimeters