Right Triangles
CH 8 Right Triangles
Radical Review
1. Simplifying
[pic]
2. Addition
[pic]
3. Multiplication
[pic]
4. Division: Rationalize the denominator
[pic]
8-1 Similarity in Right Triangles
Take out your supplementary problem
where you found the ratios in the similar triangles.
If we label the sides with the words in regards to the
larger triangle, let’s write the ratios in words.
= =
Geometric Mean: If a, b and x are positive numbers, with [pic],
then x is the geometric mean between a and b.
Ex: If NB = 16 and AN = 9, find CN, AB, BC and AC
Ex. If CN = 2 and NB = 4, find CA, AN, CB, AB
EX.
Lets simplify some more radicals:
[pic]
ALGEBRA REVIEW HOMEWORK; Also do P. 288 #1-15
8.2 The Pythagorean Theorem
Theorem 8-2 In a right triangle, the square of the ______________ is equal to the ______ of
the _________ of the legs.
What shortcuts did you discover after last night’s homework?
Remember the families that occur often?
Find the missing sides
Ex.
4)
Given isosceles trapezoid, find x
5)
6) Given the isosceles trapezoid find the altitude to the base.
7) The perimeter of a rhombus is 40 cm, and one diagonal is 12 cm long.
How long is the other diagonal?
8)
PROOFS OF PYTHAGOREAN TH Archaeology, distance formula
8.3 The Converse of the Pythagorean Theorem
Theorem 8-3 If the square of one side of a triangle is equal to the _____ ___ ____ _______
of the other two sides, then the_________ is a __________ _________
The next two theorems allow you to determine what kind of triangle you have if it isn’t a right triangle.
Theorem 8-4 If c2 < a2 + b2, then m(C < 90 and (ABC is ___________
Theorem 8-5 If c2 > a2 + b2, then m(C > 90 and (ABC is ___________
Decide the type of triangle, given the sides
1. 20, 21, 30
2. 20, 21, 28
3. 5, 6, 8
4. 5, 12, 14
5. 6, 7, 8
8-4 Special Right Triangles MEMORIZE THESE ASAP!!
Theorem 8-6 In a 45(-45(-90( right triangle, the hypotenuse is ____ times as long as
a ______
Label the following:
Theorem 8-7 In a 30(-60(-90( right triangle, the hypotenuse is _______ as long as the
shorter leg, and the longer leg is _______ times as long as the shorter leg
Label the following:
Complete the following, given the picture.
1. If x = 6 then z = ______
2. If x = [pic] then z= _____
3. If z = [pic]then y = _____
4. If z = 10 then x = _____
1. If q = 8 then p = ______ and n = ______
2. If n = 20 then q = _____ and p = ______
3. If p = [pic]then q = ______ and n = ______
4. If p = 9 then q = ______ and n = ______
5. The diagonal of a square has length 6. What is the perimeter of the square??
6. The parallel sides of trapezoid ABCD are AD and BC. Given the sides AB,BC, and CD are each half as long as Side AD, find the size of angle D.
7. What are the angle sizes in a trapezoid whose sides have lengths 6, 20, 6, and 26?
Mini-Review
1. Find the geometric mean between 5 and 30.
2. Use the diagram to fill in:
x= ______
y = ______
z = ______
The sides of a triangle are given. Is the triangle acute, right or obtuse??
3. 3, 6, 7 4. 50, 120, 130 5. 8, 10, 13
6. A rectangle has length 6 and width 2. How long is each diagonal??
7. Find the perimeter of a square if each diagonal is 8 cm long.
8. The perimeter of an equilateral triangle is 30 cm. Find the length of an altitude (now, what is an altitude???)
9. An isosceles triangle has sides 10, 10 and 12. How long is the altitude to the base??
Chapter 8 Review Right Triangles
(1) Solve for x given each of the right triangles below, with sides as indicated.
(a) (b)
(2) Given the figure below, [pic], [pic], find the indicated lengths given sides as indicated.
(a) BC = 2 , CD = 8 Find: AB_____ AC_____AD_____
(b) AD = 20, AC = 12 Find: AB_____BC_____CD_____
(c) AC = 8 , CD = 16 Find: AB_____BC_____AD_____
(d) AB = 6 , CD = 5 Find: BC_____AC_____AD_____
(3) (4)
Given: [pic], sides and angles as marked. Given: sides and angles as marked.
Find: x______ , y______ , z_______ Find: x______. Is (A obtuse, right, or acute?
(5) The perimeter of a rhombus is 20. The length of the longer diagonal is 8. How long is the shorter diagonal?
(6) The altitude of an equilateral triangle is 12. Find the perimeter of the triangle.
(7) (8)
Given: sides and angles as marked The corners are cut off a 6 inch square to form
Find: AB________ , AD________ a regular octagon as shown. Find: x______
(9) (10)
Given: sides and angles as marked Given: sides and angles as marked
Find: x______ , y______ , z_______ Find: BC________
(11) (12)
Given: [pic], sides and angles as marked Given: [pic], sides as marked
Find: AD________ and CD____________. Find: x________
Ch 8 Review Answers
(1) (a) x = 26 (b) x = [pic]
(2) (a) AB = [pic] , AC = 4 , AD = [pic]
(b) AB = 15 , BC = 9 , CD = 16
(c) AB = [pic] , BC = 4 , AD = [pic]
(d) BC = 4 , AC = [pic] , AD = [pic]
(3) x = [pic] , y = [pic] , z = [pic]
(4) x = [pic] , (A is obtuse
(5) 6
(6) [pic]
(7) AB = 13 , AD =[pic]
(8) [pic]
(9) x = 10 , y = [pic] , z = [pic]
(10) BC = 17
(11) AD = 8 , CD =[pic]
(12) x = 24
ADDITIONAL REVIEW FOR RIGHT TRIANGLES I
1. Find x, y, z
2. Isosceles trapezoid with altitude of 6.
Find the perimeter of the trapezoid.
3.
Find x, y
4. Find x, y
Answers
1. x=[pic], y=[pic], z=[pic]
2. [pic]
3. [pic]
4. [pic] [pic]
ADDITIONAL REVIEW CH 8
RIGHT TRIANGLES II
1) Find the perimeter of the isosceles trapezoid.
2) Given: [pic] sides as marked.
Find: x, y, z
3) Given: [pic], sides as marked. Find: x_____, y_____, z_____
4) Given: [pic]
Find: x_____, y_____, z_____.
5) Given: [pic], sides as marked. Find: x_____, y_____,
6) Given: trapezoid with sides as marked. Find: x_____, y_____.
7) Given: Three right triangles with three congruent angles.
Find: v_____, w_____, x_____, y_____, z_____
8) Given: If the corners are cut off a 10” square to form a regular octagon as shown, find x_____.
9) A rhombus has a [pic] angle and a 12” side. How long are it’s diagonals?
10) Find the perimeter of a square if one of it’s diagonals is [pic].
11) Given: figure as marked. Find: x_____, y_____.
12) Given: [pic]BD = CD, sides as marked.
Find: x_____, y_____, z_____.
13) Given: [pic], sides as marked.
Find: x_____, y_____.
14) Given: [pic], sides as marked. Find x_____, y_____, z_____.
Ch 8 Additional Review II Answers
1) P = 38
2) x = 36, y = 34
3) x = 12, y = [pic], z = [pic]
4) x = [pic]
5) [pic]
6) [pic]
7) [pic]
8) [pic]
9) [pic]
10) [pic]
11) [pic]
12) [pic]
13) [pic]
14) [pic]
SUPPLEMENTARTY PROBLEMS CH 8
1. Triangle ABC is a right triangle with right angle at C and an altitude from C to the hypotenuse, [pic].
Name three similar triangles and set up 3 ratios for each triangle duo. There should be 3 combinations.
Ex. [pic]_ _ _ [pic][pic]_ _ _
Give three proportions for these similar triangles
[pic]_ _ _ [pic][pic]_ _ _
Give three proportions for the next group of similar triangles
[pic]_ _ _ [pic][pic]_ _ _
Give three proportions for the third grouping of similar triangles
2. In baseball, the bases are placed at the corners of a square whose sides are 90 feet long. Home plate and second base are at opposite corners. How far is it from home plate to second base? You may leave your answer in radical form.
3. One of the legs of a right triangle is twice as long as the other and the perimeter of the triangle is 28. Find the length of all three sides.
4. Confused after grading many journals, Mrs. McGrath walked one mile due north, two miles due east, then three miles due north again, and then once more east for 4 miles. How far is Mrs. McGrath from her starting point?
5. Max and Mark were out in their rowboat one day, and Max spied a water lily. Knowing that Mark liked a mathematical challenge, Max announced that, with the help of the plant, it was possible to calculate the depth of the water under the boat. When pulled taut, the top of the plant was originally 10 inches above the water surface. When Mark help the top of the plant, which remained rooted to the lake bottom, Max gently rowed the boat 5 feet. This forced Mark’s hand to the water surface. Use this information to calculate the depth of the water.
[pic] [pic]
6. A triangle with sides 5,12 and 13 must be a right triangle. Keeping the legs constant, how would the triangle change if the hypotenuse was lengthened to 15? To 17? 19?
7. The diagonals of a square have length 10. How long are the sides of the square?
What is the ratio of the side to the diagonal?
8. The legs of an isosceles right triangle have the length “s”. What is the length of the hypotenuse with respect to s?
9. An equilateral triangle has a side of length “s”. Find the altitude to a side in terms of “s”.
10. What is the length of a side of an equilateral triangle whose altitude is 16?
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x
45
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“a”
“a”
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2
C
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Segment 2
Segment 1
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m-2
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hypotenuse
alt
leg 2
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5
leg 1
15
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[pic]4
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[pic]
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Hint: Look for shortcuts
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