8.2-apply similiarity concepts, scale factors and proportional reasoning to solve measurement problems

what is the scale factor, k...?

1. 1944=k³×9 1944/9=216, the scale factor is 6.

P. 570- # 107 - If both the length and the width of a rectangle are doubled, how many times larger is the area of the resulting rectangle...?

1. 2(l+w) 2. A= 4lw, four times larger

figure 1: 20000mi³ and figure 2: 10240mi³, what is the scale factor...?

1. 20000=k³×10240, divide by 10240 on both sides 2. Solve for k³ 3. 5:4 final answer

Scale factor = 4:9, SA= 256km², V=1536km³, what is the new SA & V?

1. 256=k²×x; 1536=k³×y 2. 256=(4/9)²×X; 1536=(4/9)³×Y 3. 256=16/81x, 81×256=16x; 1536=64/729×y, 729×1536=64y 4. 20736/16=x, 1296km²; 1119744/64=17496km³

11x-4/70=60/50, solve for x...?

1. 50(11x-4)=60(70) 550x-200=4200 +200. 200 4400 X=8

what is the new SA?, Current SA= 54m², k= 3/7...

1. 54=k²×y 2. 54=(3/7)²×y 3. 54=9/49×y 4. 54×49=9y, 294m²

Scale factor = 1:2, SA= 90yd², V=216yd³, what is the new SA & V?

1. 90=(½)²×X, x=360 2. 216=(½)³×Y, y=17284 360yd² and 17284yd³

Draw a line parallel to a b through C. Let G be the intersection of the new line and the angle bisector, proof...?

1. <ABD= <CED 2. TRIANGLE BCE is an isosceles triangle, and hence why BC=CE 3. <BDA=<CDE 4. △ABD𝀈△CED 5. AB/AD=CE/CD 6. AB/AD=BC/CD

△TUS, xxT, xxxU, △BSC...xB, x= tick marks..., triangles similar?

1. <CSB≅<TSU due to vertical angles 2. sides are not congruent, therefore triangles are not similar

P. 570 - # 108 - Find the ratio of the areas of two squares if the ratio of the lengths of their sides is 2:3.

1. A₁=(2x)² and A₂=(3x)² 2. 4x²/9x²...4/9 ​

Suppose a circle is cut into 16 equal pieces, which are then arranged as shown in the attached. The figure formed resembles a parallelogram. What verbal expression could describe the base of the parallelogram? what variable could describe its height? explain how the formula for the area of a circle derived from this approach.

1. b=½(2πr), = πr 2. h=√(r²-(½×πr/8)² 3. √(r²-π²r²/16²), = r/16√(16²-π²) => r/16√(256-9.87) => r/16×15.69 => .98r 4. The area of the circle is nearly = the area of the parallelogram.

The dimensions of the smaller cylinder are two thirds of the larger. The volume of the larger cylinder is 2160 п cm^3. Find the volume of the smaller cylinder.

2^3/3^3 = V/2160п

√(25/4) = k², what is the scale factor...?

5/2

Another way to represent the attached tiles.

8N+4=(3N+1)+(3N+1)+(2N+1)

Solve the proportion, . x/x+3 = 34/40

= 34x+ 102 = 40x = -6x = -102 = -6x = -102, x = 17

P. 568 - #88- A fabric wall hanging is to fill a space that measures 5 m by 3.5 m. Allowing .1 m of the fabric to be folded back along each edge, how much fabric must be purchased for the wall hanging?

=(5+2×.1)×(3.5+2×.1) =5.2×3.7m² =19.24m²

Thales of Miletus (THAY-lees)

A Greek natural philosopher (ca. 624 - 547 BCE), noted for his application of reason to astronomy and for his questioning of the fundamental nature of the universe. He discovered that he could determine the heights of pyramids and other objects by measuring a small object in the length of its shadow and then making you to the similar triangles.

P. 570- #103-A walkway 2 m wide surrounds a rectangular plot of grass. The plot is 30 m long and 20 m wide. What is the area of the walkway?

A=(20+2+2)×(30+2+2)m² =24×34m² ∴(816-600)m² =216m²

8.2- #59- he scale on a topographical map is 5/4 of an inch equals one mile. The distance between 2 mountain peaks on the map is 4⅛ inches. How many miles apart of those 2 peaks?

Competency 8 this question requires examinees to apply concepts of similarity, scale factoring and proportional reasoning to solve measurement problems.. 1. 5/4= 1mi 2. ⅘= 33/4 3. 6.6 miles

8.2 - A company sells dog food In metal cans of 2 sizes: large and giant. The cans are 6" and 9" tall, respectively, of uniform thickness, and similar in shape. If 8 cents worth of metal is used to make the large can, what is the value of the Metal used to make the giant can? Unitary method

Competency 8 this question requires the examinee to apply the concepts of similarity, scale factors and proportional reasoning to solve measurement problems of measurement problems. 1. Size of large cans = 6" and size of giant cans= 9" 2. Cost of metal to make large cans is 8 cents cost of metal to make giant cans is ? 3. 6ft can us 8 cents= 8/6 and 9ft can = cost?; 8/6=9/1=> 12 cents

corresponding and similar triangles

Congruent and similar triangles differ in that the corresponding sides and angles of congruent triangles must be equal, whereas for similar triangles, corresponding angles are equal but corresponding sides are not necessarily the same length

Side-Side-Side (SSS)

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Angle-Angle Similarity Postulate

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

If the four angles of one quadrilateral are congruent to the four corresponding angles of another quadrilateral, then the two quadrilaterals are similar.

False, one possible counterexample is attached.

If two sides of one triangle are proportional to the sides of another triangle, then the two triangles are similar.

False, one possible counterexample is shown.

Dilation Theorem

If a dilation with center C and scale factor of r transforms D to A and B to E, then AE = |r|(BD)

Triangle Proportionality Theorem

If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.

Angle Bisector Theorem

If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle

Side-Angle-Side Similarity Theorem

If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.

Triangle ABC is similar to Triangle DBA X = ?, y = ?

Redraw the two triangles so that the corresponding angles are in the same position. 6/5 = 5/x and 9/6 = y/5

SA=7πin², SA=175πin², scale factor...?

SA=7= 7×1 SA=175=7×5×5 1:5

Similar Triangles

Similar triangles have the same shape: corresponding angles are equal and corresponding sides are proportional

The Jones family paid $150 to a painting contractor to stain their 12 ft by 15ft back deck. The Smiths, their neighbors have a similar deck that measures 16ft by 20ft. If the Smiths wish to "keep up with the Joneses." What is a proportional price the Smiths family should expect to pay to have their deck stained by the contractor?

The fact that the two decks are similar rectangles is irrelevant. The painting contractor charges by time or square footage, and thus $150/ 12x15 = 5x/16 x20

corollary of proportionality Theorem

a transversal that is parallel to a side in a triangle defines a new smaller triangle that is similar to the original triangle

how do you show that corresponding angles in 2 triangles are congruent (equal)?

a+b/a=c+d/c=AB/DE

what do you need to prove the triangle proportionality Theorem...?

area of a triangle, A=w×h/2

scale factor

new/original, in two similar geometric figures, the ratio of their corresponding sides

2012≠2200, sides prortional...yes or no?

no, sides need to equal one another

Dilation with a scale factor r>1

produces an image larger in size, but the angles of the figure are unchanged and the lengths of the larger figure are proportional with the original figure.

a dilation with a scale factor =1...

produces an image that is congruent to the original figure

dilation with a scale factor, 0<r<1...

produces an image that is smaller in size in the original, but the angles of the figure unchanged, and the length of the smaller figure are proportional with the original figure

a figure is transformed under a dilation with a scale factor of?

r=1, r>1, and 0<r<1...

what is the ratio of poly3 and poly2..., how do you prove?

show that a/b=c/d

Solve the proportion, 4/x = x/9

x^2 = 36, x = ±6.244

proportional, 10/9=27/30 yes or no....?

yes, 10/9=27/30, 270=270...

proportional, 9/18=21/42 yes or no....?

yes. 9/18=21/42, 378=378

properties of similar triangles

They have the same shape, but not necessarily the same size. Their corresponding angles are equal and their corresponding sides are proportional. the ratio of corresponding heights is equal to the ratio of corresponding sides.

The Ring a Ding Sisters Circus has come to town. P.T. Barnone is the star of the show. She does a juggling act atop a stool that sits atop of a rotating ball that spins at the top of a 20-meter pole. The diameter of the ball is 4 meters, and P.T.'s eye is 2 meters above the ball. The circus manager needs to know the radius of the circular region beneath the ball in which spectators would be unable to see eye to eye with P.T. Find the radius to the nearest tenth meter for the manager so that he can put in the seats for the show. (Use 1.7 for square root 3)

Triangle ABE is similar to triangle ADC AB/BE = AD/CD

If a line parallel to one side of a triangle passes through the other two sides, then it divides them proportionally.

True, if a line divides two sides of a triangle proportionally, then it is parallel to the third side.

Vertical Angles Theorem

Vertical angles are congruent

Transversal

a line that intersects two or more lines