Question 20 - End-of-Chapter Exercises - Chapter 6 Class 9 - Measuring Space: Perimeter and Area (Ganita Manjar

Transcript

Question 20 Show that the areas of the shaded blue triangle and the shaded red triangle are equal. Find a way of cutting up the blue triangle into some number of pieces and rearranging the pieces to cover the red triangle. Since base of big triangle is trisected Length of base in grey, white and red triangle is the same Let’s first show that Areas are equal tep 1 of 5 The Problem We have a large triangle. Lines are drawn from the top vertex to the points of trisection on the opposite side. Show that the Blue and Red triangles have equal area. Step 2 of 5 Base and Height Because they are trisection points, the base is cut into 3 exactly equal pieces. Let's call each base . Notice that both the Blue and Red triangles reach up to the exact same top vertex. This means they share the exact same height (h).Step 3 of 5 Equal Area Proof The area of any triangle is 1/2× base × height. Blue Area =1/2×h Red Area =1/2×h Mathematically, their areas are identically equal! Now, we need to answer Find a way of cutting up the blue triangle into some number of pieces and rearranging the pieces to cover the red triangle. Because they have the exact same base and height, geometry guarantees we can cut one up to build the other. Here is a simple way to do it: Cut the blue triangle horizontally exactly halfway up its height. Cut the top small blue triangle straight down the middle. Because the red triangle leans further to the right, you can take those two top pieces from the blue triangle and slide them down the slanted edge of the bottom piece to rebuild the rightward-leaning red triangle! Step 4 of 5 The Slicing Method Because they have the exact same base and height, geometry guarantees we can cut one up to build the other. Here is a simple way to do it: Cut the blue triangle horizontally exactly halfway up its height. Cut the top small blue triangle straight down the middle.Step 5 of 5 Rebuilding the Triangle Because the red triangle leans further to the right, you can take those two top pieces from the blue triangle and slide them down the slanted edge of the bottom piece to rebuild the rightwardleaning red triangle!