Ratios/Proportions
Similar Polygons
Proving Triangles Similar
Right Triangles
Proportional Segments
100
Use the Means-Extremes Property to solve:
(x + 1)/9 = (x - 3)/3
What is x = 5.
3(x + 1) = 9(x - 3)
3x + 3 = 9x - 27
30 = 6x
x = 5
100
True or False:
Similar polygons are also congruent polygons.
What is False.
100
What are the three methods for proving triangles similar?
What are AA, SAS~ and SSS~
100
In a triangle, if c2 > a2 + b2, then what kind of triangle is it?
What is an obtuse triangle.
100
Given triangle ABC, and ray AD bisects angle A. What proportion can be made relating the lengths of the sides of angle A to the segments formed in side BC?
What is AB/AC = BD/DC or AB/BD = AC/DC. (Theorem 5.6.3, The Angle-Bisector Theorem)
200
The measures of the three interior angles of a triangle are in the ratio of 4:8:12.
Classify the triangle based on its angles.
What is a right triangle (30, 60, 90 right triangle).
4x + 8x + 12x = 180
24x = 180
x = 7.5
200
True or False:
Two polygons are similar if each angle of one polygon is congruent to an angle of the other polygon.
What is False.
They have to be corresponding vertices.
200
There are two triangles ABC and DEF. If angle A is congruent to angle D, and AB/DE = AC/DF, can we use this to prove the triangles similar?
What is yes by SAS~.
Angles A and D are the included angles of the proportional sides.
200
One side of a rhombus has the same length (10 in) as each side. How long is the other diagonal?
What is the square root of 75.
In a rhombus, the diagonals are perpendicular bisectors of each other, so four right triangles are formed.
52 + b2 = 102
25 + b2 = 100
b2 = 75
200
The altitude drawn to the hypotenuse of a right triangle separates the right triangle into . . .?
What is two right triangles that are similar to each other and to the original right triangle.
300
If a/b = c/d (where b and d are not 0),
what is (a + b)/b equal to?
What is (c + d)/d.
300
What are two conditions necessary to say two polygons are similar?
What is all pairs of corresponding angles are congruent, and all pairs of corresponding sides are proportional.
300
Can I use CSSTP or CASTC to prove triangles are similar?
What is no. These can only be used after you have proved triangles are similar.
300
Find the perimeter of a square if a diagonal has a length of 3 times the square root of 2.
What is 12.
A diagonal divides the square into two isosceles right triangles.
If the diagonal (hypotenuse) is 3 times the square root of 2, then each leg of the triangle is 3,
which is the length of one side of the square.
300
The length of the altitude to the hypotenuse of a right triangle is the geometric mean of . . .?
What is the lengths of the segments of the hypotenuse (Theorem 5.4.2)
400
Assume that AD is the geometric mean of BD and BC.
If AD = 6 and DC = 8, find BD.
What is 4.5.
BD/AD = AD/BC
8x = 36
x = 4.5
400
I am curious as to how high the APC building is. On a sunny day, my shadow is 9 feet, and the shadow of APC is 60 feet. If I am 6 feet tall, how high is APC?
What is 40 feet.
Using similar triangles, you get the proportion
6/9 = h/60
9h = 360
h = 40
400
Daily Double!
See the figure on the board. (#18, P 273)
Given: Angle 1 is congruent to angle 2
Prove: AB/AC = BE/CD
What is:
Angle 2 is congruent to angle ADC (vertical angles)
Angle A is congruent to itself (Reflexive Prop)
Triangle ACD is similar to triangle ABE (AA)
AB/AC = BE/CD (CSSTP)
400
Given rectangle HJKL with diagonals HK and JL.
The measure of angle HKL is 30 degrees,
and KL = 6 times the square root of 3.
Find HL and HK.
What is HL = 6 and HK = 12.
The diagonal forms a 30, 60, 90 right triangle with the long leg 6 times the square root of 3.
So, the shorter leg is 6 and the hypotenuse is 12.
400
The length of each leg of a right triangle is the geometric mean of . . .?
What is the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to that leg. (Theorem 5.4.3)
500
If the ratio of the measure of the complement of an angle to the measure of its supplement is 1:6, find the measure of the angle.
What is 72 degrees.
(90 - x)/(180 - x) = 1/6
180 - x = 6(90 - x)
180 - x = 540 - 6x
5x = 360
x = 72
500
Are any two isosceles triangles similar?
Any two rhombuses?
Any two circles?
What is No (isosceles triangles), No (rhombuses), Yes (circles).
500
If a line segment divides two sides of a triangle proportionally, then what is the relationship of this line segment to the third side of the triangle?
What is parallel (Lemma 5.3.5)
500
A triangle has sides of lengths 4, 5, and 6.
Find or describe how to find the length of the altitude to the side of length 6.
What is (the square root of 175 )/4. (approx. 3.31)
The altitude to the side of length 6 separates that side into two parts whose lengths are x and 6 - x.
Letting the length of the altitude be h, we apply the Pythagorean Theorem twice to get:
x2 + h2 = 42
(6-x)2 + h2 = 52
Subtracting the first equation from the second we can get x.
36 - 12x = 9
27 = 12x
x = 27/12 = 9/4
Substituting into the first equation we get:
(9/4)2 + h2 = 42
81/16 + h2 = 256/16
h2 = 175/16
h = (the square root of 175)/4
500
Given parallel lines EF, GO, HM, and JK with transversals FJ and EK. if FG = 2, GH = 8 HJ = 5 and EM = 6, find EO andEK.
What is EO = 6/5 and EK = 9
First find EO
FH/EM = FG/EO, so 10/6 = 2/EO
10(EO) = 12
EO = 12/10 = 6/5
Now find EK
FJ/EK = FH/EM
15/EK = 10/6
10(EK) = 90
EK = 9
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