Topic 1 Pythagorean Theorem

Transcription

1 Topic 1 Pythagorean Theorem Question 1 name Perimeter of a trapezoid text In trapezoid ABCD the sides AB and CD are equal. The lengths of BC and AD are b 1 = $x and b 2 = $y respectively. The height of the trapezoid is h = $z. Determine the perimeter of ABCD. answer $ans 1

2 code $a = int( rint(3) ); $x = int( rint(5) ) + 3; $z = int( $a*3); $y = int( $x + $a*2*4 ); $ans = int( $x + $y + $a*5); comment Adapted from 1999 AMC 8, Problem 14, 25.65% Question 2 name Thirds of a square text Square ABCD has sides of length $x. Segments CM and CN divide the square s area into three equal parts. How long is segment CM? answer $ans code $x = int( rint(5) ) + 3; $ans = sqrt( (13/9)*x^2 ); comment Adapted from 1999 AMC 8, Problem 23, 17.98% Question 3 name Overlapping Squares text Two $a $a squares intersect at right angles, bisecting their intersecting sides, as shown. The circle s diameter is the segment between the two points of intersection. What is the area of the shaded region created by removing the circle from the squares? 2

3 answer $ans code $a = int( rint(5) ) + 3; $ans = 7*a^2/4 - a^2*pi/8 ; comment Adapted from 2004 AMC 8 Problem 25, 13.12% correct Question 4 name Pythagorean Theorem and the perimeter of a trapezoid text The area of trapezoid ABCD is $area cm 2. The altitude is h = $height cm, AB is r = $lefthyp cm, and CD is s = $righthyp cm. What is BC, in centimeters? answer $upperbase code $r = rint(4); $height = switch($r, 8, 12, 24, 16); $lefthyp = switch($r, 10, 13, 25, 20); $leftleg = switch($r, 6, 5, 7, 12); 3

4 $righthyp = switch($r, 17, 20, 26, 34); $rightleg = switch($r, 15, 16, 10, 30); $upperbase = rint(3) + 10; $lowerbase = $leftleg + $upperbase + $rightleg; $averagebase = ($upperbase + $lowerbase)/2; $area = $height*$averagebase; comment Adapted from 2003 AMC 8 Problem % correct Question 5 name Area from the Pythagorean Theorem text Given the areas C = $c2 and A = $a2 of the two squares in the figure, what is the area of the interior triangle? answer $area code $r = rint(8); $a = switch($r, 3, 5, 6, 7, 8, 10, 12, 16); $b = switch($r, 4, 12, 8, 24, 15, 24, 16, 30); $c = switch($r, 5, 13, 10, 25, 17, 26, 20, 34); $a2 = $a^2; $c2 = $a^2; $area = (1/2)*$a*$b; comment Adapted from 2003 AMC 8 Problem 6, 47.69% correct Question 6 mode Multiple Choice name A triangle view of the Pythagorean Theorem text Right isosceles triangles are constructed on the sides of a triangle, as shown. A capital letter represents the area of each triangle. Which one of the following is true? 4

5 choice X + Z = W + Y choice W + X = Z choice 3X + 4Y = 5Z choice X + W = 1 (Y + Z) 2 correct-choice X + Y = Z comment 2002 AMC 8 Problem % correct Question 7 name Rectangle into three triangles text In rectangle ABCD, AD = $ad, P is on AB, and DB and DP trisect What is the perimeter of BDP? ADC. answer $p code $ad = rint(10); 5

6 $db = 2*$ad; $ab = sqrt3*$ad; $ap = (sqrt3/3)*$ap; $dp = 2*(sqrt3/3)*$ap; $pb = $ab - $ap; comment Adapted from 2000 AMC 10, Problem 7, 77.28% Question 8 name Altitude of a triangle text The sides of a triangle have lengths of $a, $b and $c. Find the length of the shortest altitude. answer $alt code $r = rint(10); $a = 3*$r; $b = 4*$r; $c = 5*$r; comment Adapted from 2002 AMC 10 A, Problem 13, 16.06% Question 9 name Size of a television screen text Many television screens are rectangles that are measured by the length of their diagonals. Likewise, for many television screens the ratio of the horizontal length to the height in a standard television screen is 4 : 3. The horizontal length of a $d-inch television screen conforming to these standards is closest to which of the following? answer $a code $r = rint(5); $d = switch($r, 20, 27, 32, 34, 37); $a = round( (5/4) * $d ); comment Adapted from 2003 AMC 10 B, Problem 6, 38.44% correct Question 10 name Distance between cities text Suppose the Regional Airport is $a miles southwest the center of City A and $b miles southeast of the center of City B. Which of the following is closest to the number of miles between the centers of City A and City B? answer $c code $r = rint(10) + 2; $a = $r*4; $b = $r*5; $c = round( sqrt(41)*$r ); comment Adapted from 2004 AMC 10 B, Problem 8, 63.25% correct 6

7 Question 11 name Perimeter of another trapezoid text In the trapezoid ABCD, AD = $lefthyp, DC = $50, CD = $righthyp. The height DE = $height. What is the perimeter of trapezoid ABCD? answer varperimeter code $r = rint(4); $height = switch($r, 8, 12, 24, 16); $lefthyp = switch($r, 10, 13, 25, 20); $leftleg = switch($r, 6, 5, 7, 12); $righthyp = switch($r, 17, 20, 26, 34); $rightleg = switch($r, 15, 16, 10, 30); $upperbase = 50; $lowerbase = $leftleg + $upperbase + $rightleg; $perimeter = $upperbase + $lowerbase + $lefthyp + $righthyp; comment Adapted from 2005 AMC 8, Problem 19, 38.75% correct Question 12 7

8 name Bill takes a walk text Bill walks $a mile south, then $b mile east, and finally $a mile south. How many miles is he, in a direct line, from his starting point? answer $a code $r = rint(5); $a = (1/2)*$r; $b = (3/4)*$r; $c = (5/4)*$r; comment Adapted from 2005 AMC 8, Problem 7, 22.26% correct 8