Honors Geometry Chapter 7 Test
P(4, 7) →(4 × 6, 7 × 6) → P'(24, 42) Q(11, 18) → (11 × 6, 18 × 6) →Q'(66, 108) P'Q' = √(66-24)² + (108-42)² P'Q' = √42² + 66² P'Q' = √1764 + 4356 = √6120 P'Q' ≈ 78.2
PQ with endpoints P(4, 7) and Q(11, 18) is dilated by a scale factor of 6. Find the length of P'Q' to the nearest tenth.
ratio
a comparison of two quantities by division
scale drawing
a drawing that uses a scale to represent an object as smaller or larger than the original object
1cm/5ft = w/45ft 5w = 45 w = 9ft 1cm/5ft = ℓ/70ft 5ℓ = 70 ℓ = 14ft
a house is 45 feet wide and 70 feet long. if a sketch is made of the house using the scale 1 cm: 5 ft, what are the dimensions of the sketch?
indirect measurement
a method of measurement that uses formulas, simular figures, and/or proportions
1in/16ft = 13in/xft x = 208 ft
a model airplane is built to a scale of 1 in.:16 ft. if the model plane is 13 inches long, find the length of the actual plane, rounded to the nearest foot.
proportion
a statement that two ratios are equal
dilation
a transformation in which the lines connecting very point P with its preimage P' all intersect at a point C known as the center of dilation, and CP'/CP is the same for every point P; a transformation that changes the size of a figure but not its shape.
20/24 = 50/x 20x = 1250 x = 60 ft
a tree is standing next to a 50-foot high building. the tree has an 24-foot shadow, while the building has a 20-foot shadow. how tall is the tree, rounded to the nearest foot?
340/1 = 1/250 =1.36 ft = 16.32 in height = 16.32 inches 270/1 = 1/250 =1.08 ft = 12.96 in width = 12.96 inches
a video game designer is modeling a tower that is 340 ft high and 270 ft wide. she creates a model so that the similarity ratio of the model to the tower is 1/250. what is the height and the width of the model in inches?
proportional
an angle bisector of a triangle divides the opposire side into two segments whose lengths are _________________ to the lengths of the other two sides (triangle angle bisector theorem)
AD ‖ BF ‖ CH AB/DF = BC/FH 5/4 = x/3 4x = 15 BC = 3.75 cm
an artist used perspective to draw guidelines in her picture of a row of parallel buildings. how many centimeters is it from Point B to Point C?
a = (½b) x h
area of a triangle
12/96 = 18/x 12x = 1728 x = 144 in
coby designs a rectangular vegetable garden. what will be the length of the full-size vegetable garden?
all ∠s rt ∠s, so all ∠s ≅ AB/MN = 39/26 = 3/2 DC/LP = 39/26 = 3/2 BC/LM = 18/12 = 3/2 AD/NP = 18/12 = 3/2 the similarity ratio is 3/2 and rect. ABCD ∼ rect. LPNM
determine whether the rectangles are similar. if so, write the similarity ratio and a similarity statement from the left rectangle to the right rectangle.
d = √(x₂-x₁)² + (y₂-y₁)²
distance formula
∠C ≅ ∠C by reflexive property ∠EAC ≅ ∠DBC by corresponding ∠s postulate ∆EAC ∼ ∆DBC by AA∼
explain why the triangles are similar and write a similarity statement.
10/w+2 = 8/2w 2w(10) = 8(w+2) 20w = 8w + 16 12w + 16 w = 4/3 BD = w + 2 + 2w BD = 4/3 + 2 + 2(4/3) BD = 4/3 + 2 + 8/3 BD = 12/3 + 2 = 4 + 2 = 6
find BD
QS/SP = RT/TP 2/6 = x/9 6x = 18 x = 3 RT = 3
find RT.
m/n = 12÷4/16÷4 m/n = 3/4
given that 12m = 16n, find the ratio of m to n in simplest form.
m = 7-12/11-5 m = -5/6
given that two points on line m are P(5, 12) and Q(11, 7), write a ratio expressing the slope of m.
since ∆KON ∼ ∆LOM, NO/MO = KO/LO -4/-12 = 3/x 4x = 36 x = 9 L = (9, 0) (3, 0) →(3 × 3, 0 × 3) →(9, 0) the scale factor is 3
given that ∆KON ∼ ∆LOM, find the coordinates of L and the scale factor
the similarity ratio is 13/9.1 the ratio of the perimeters is 13/9.1 the ratio of the areas is (13/9.1)² perimeter: 36/P = 13/9.1 13P = 327.6 P = 25.2 cm area: 60/A = (13/9.1)² 13²A = (9.1)² (60) A = 29.4 cm²
given ∆LMN ∼ ∆QRS, find the perimeter and area of ∆QRS
∠L and ∠I ∠F and ∠H ∠O and ∠T FL/HI = 15/30 = 1/2 LO/IT = 20/40 = 1/2 FO/HT = 25/50 = 1/2 the similarity ratio is 1/2
identify the pairs of congruent angles and corresponding sides. Write the corresponding sides as ratios from the left triangle to the right triangle, and find the similarity ratio.
4/10 = 20/x 4x = 200 x = 50
if 4, 12, and 20, and 10, 30, and x are the lengths of the corresponding sides of two similar triangles, what is the value of x?
parallel
if a line divides two sides of a triangle proportionally, then it is _________________ to the third side (converse of the triangle proportionality theorem)
proportionally
if a line parallel to a side of a triangle intersects the other two sides, then it divides those sides _________________ (triangle proportionality theorem)
a/b; (a/b)²
if the similarity ratio of two similar figures is a/b, then the ratio of their perimeters is ____ and the ration of their areas is ____ (proportional perimeters and areas theorem)
proportional
if the three sides of one triangle are _________________ to the three corresponding sides of another triangle, then the triangles are similar (side-side-side similarity theorem)
proportionally
if three or more parallel lines intersect two transversals, then they divide the transversals _________________ (two-transversal proportionality)
congruent
if two angles of one triangle are _________________ to two angles of another triangle, then the triangles are similar (angle-angle similarity postulate)
proportional; congruent
if two sides of one triangle are _________________ to two sides of another triangle and their included angles are _________________, then the triangles are similar (side-angle-side similarity theorem0
extremes
in the proportion a/b=c/d, "a" and "d" are the __________. if the proportion is written as a:b=c:d, then the ___________ are the first and last position
means
in the proportion, a/b=c/d, "b" and "c" are the ________
cross products
in the statement a/b = c/d, "bc" and "ad" are the ___________ ______________
2.5m/1.3m = 9.5cm/x 2.5x = 12.35 x ≈ 4.9 cm
maya is making a miniature dinner table for her little sister. she wants the table top to be similar to their real dinner table top. find the width of the miniature table top to the nearest tenth of a centimeter.
(4/15)² 16/225
one equilateral triangle has sides 4 ft long. another equilateral triangle has sides 15 ft long. find the ratio of the areas of the triangles.
36w = 92 36w/36 = 92/36 w = 23/9
solve the proportion 3/4 = 23/12w
1/4 = x/16 4x = 16 x = 4 ft to belt buckle 1/4 = (7/12)/x 28/12 x = 2 1/3 ft to diameter of head
the city of bangor, maine has a scale model of paul bunyan nearly 50 feet tall. the model's scale is 1:4. on the scale model, paul bunyan's belt buckle is 16 feet from the ground. in real life, how far from the ground is paul bunyan's belt buckle? the diameter of paul bunyan's actual head is 7 inches. what is the diameter of the paul bunyan's scale model head in feet?
A(0, 0) → (0 × ½, 0 × ½) →A'(0, 0) B(0, 3) → (0 × ½, 3 × ½) → B'(0, 1.5) C(2, 3) → (2 × ½, 3 × ½) → C'(1, 1.5) D(2, 0) → (2 × ½, 0 × ½) → D' (1, 0) A'(0, 0); B'(0, 1.5); C'(1, 1.5); D'(1, 0)
the figure shows the position of a photo. what are the vertices of the photo after a dilation with scale factor ½.
scale factor
the multiplier used on each dimension to change one figure into a similar figure
scale
the ratio between two corresponsing measurements
4x + 7x + 3x + 6x = 150 m 20x = 150 m x = 7.5 m 3(7.5m) = 22.5 m
the ratio of the side lengths of a quadrilateral is 4:7:3:6, and its perimeter is 150 meters. what is the length of the shortest side?
similarity ratio
the ratio of two corresponding linear measurements in a pair of similar figures
30ft/10ft = x/20ft 10x = 600 d = 60 ft.
to find out how wide a river is, jon and sally mark an X at the spot directly across from a big rock on the other side of the river. then they walk in a straight line along the river, perpendicular to the straight line between the X and the rock. after walking for 20 feet jon stops while sally continues along the straight line for another 10 feet. then she makes a 90 degree turn and walks for 30 feet. when she stops and looks at the rock she sees that the straight line from her to the rock passes through jon. what is the distance from X to the rock?
similar
two figures are ___________ if they have the same shape but not necessarily the same size
similar polygons
two polygons whose corresponding angles are congruent and whose corresponding sides are proportional
AB/A'B' = 10/4 = 5/2 BC/B'C' = 5/2 ∠B ≅ ∠B by reflexive property ∆ABC ∼ ∆A'B'C' by SAS ∼
verify that ∆ABC ∼ ∆A'B'C'.
