Problem 350

Problem 350

A tangram is a geometric puzzle made from seven pieces that from a square. Trace the pieces, cut them out, and answer the following questions.

a. If the area of the entire square equal 1 unit, label each piece with its rational number area.

b. If the area of pieces (a) equals 1 unit, label each piece with its rational number area.

Segments marked (/) are all the same length. Segments marked (//) are all the same length.

Solution

Area (a) + area (b) = $\frac{1}{2}$

We know Area (a)= area (b)

So area (a) = $\frac{1}{2}$ ( $\frac{1}{2}$ ) = 1/4

Area (b)= $\frac{1}{2}$ ( $\frac{1}{2}$ ) = $\frac{1}{4}$

Area (g)= $\frac{1}{2}$ ( $\frac{1}{2}$ )( $\frac{1}{2}$ ) = $\frac{1}{8}$

One side of (d) = $(\frac{\sqrt{2}}{2}) = \frac{\sqrt{2}}{4}$

Area (d)= ( $\frac{\sqrt{2}}{4}$ )²= $\frac{1}{8}$

Area (c)= $\frac{1}{2}$ ( $\frac{\sqrt{2}}{4}$ )( $\frac{\sqrt{2}}{4}$ )=1/16

Area (e)= Area (c)= $\frac{1}{16}$

Area ((a)+ (b)) = area ((c) +(d) +(e) +(f) +(g) )

$\frac{1}{4}$ + $\frac{1}{4}$ = $\frac{1}{16}$ + $\frac{1}{8}$ + $\frac{1}{16}$ + area (f) + $\frac{1}{8}$

$\frac{1}{2}$ = $\frac{5}{8}$ + area (f)

Area (f) = $\frac{1}{2}$ – $\frac{3}{8}$ = $\frac{1}{8}$

Final answer a = b = $\frac{1}{4}$ ; c = e = $\frac{1}{16}$ ; d = g = f = $\frac{1}{8}$ .

b. If piece a = 1, then we can find the other pieces simply multiplying by 4. Hence,

a = b = 1; c = e = $\frac{1}{4}$ ; d = f = g = $\frac{1}{2}$