Notes #3-___
Date:______
11-2 The Pythagorean Theorem (584)
W.1Simplify each expression:
a)52 + 62b)92 – 42c)(3t)2 + (4t)2
W.2 Simplify each radical expression:
a)b)
Right triangle: a triangle with a 90° angle.
Hypotenuse: the longest side of a rtΔ (across from 90° angle).
Legs: the perpendicular sides of a right triangle
Square the length of each side. Compare the sum of two smaller numbers to the larger number. What do you see?
Pythagorean Theorem
If a triangleis a right triangle, then the sum of the squares of the lengths of the legs a and b equals the square of the length of the hypotenuse c. a2 + b2 = c2.
Ex.1Find the missing side (a, c & d Pythagorean Triples):
a)a = 8 & b = 15b)a = 2 & c = 6
c)a = 7 & b = 24d)b = 8 & c = 10
Ex.2A toy fire truck is positioned so that the base of the ladder is 13cm from the wall. The ladder is extended 28cm to the wall. The distance from the top of the table to the bottom of the ladder is 9cm. How high above the table is the top of the ladder?
Converse of the Pythagorean Theorem
If a triangle has lengths a, b and c such that a2 + b2 = c2, then the triangle is a right triangle (c must be the largest side).
Define converse:______
Ex.3Determine whether the given lengths can be sides of a
right triangle.
a)5 in, 5 in, 7 inb)10 cm, 24 cm, 26 cm
Ex.4If two forces pull at right angles to each other, the resultant force is represented as the diagonal of a rectangle. For a 50-lb force and a 120-lb force, the resultant force is 130-lb. Are the forces pulling at right angles to each other?
Ex.5A right triangle’s hypotenuse is 1 ft longer than its
longer leg. The shorter leg is 9 ft. Find the lengths.
Ex.6A television screen has sides 20 in & 13 in. How
many inches long is the diagonal?
a)b)c)33d)569
Ex.7Find the missing side. What kind of triangles are a-d?
a)e)
b)f)
c)g)
d)h)
Ex.8A 8 m longladder is resting against a building. The
bottom of the ladder is 3 m from the wall. How far up
the wall does the ladder reach?
Ex.9Which of the following line segments (not shown)
has the greatest length?
a)
b)
c)
d)
e)
Ex.11The lengths of the sides of a right triangle are
consecutive even integers and the length of the
shortest side is x. Which of the following equations
could be used to find x?
a)x + x + 1 = x + 2b)x2 + (x + 1)2 = (x + 2)2
c)x + x + 2 = x + 4d)x2 + (x + 2)2 = (x + 4)2
e)x2 = (x + 2)(x + 4)
Notes #3-___
Date:______
11-3 The Distance and Midpoint Formulas (393)
W.1Find the missing lengths in the right triangle if the
lengths of the legs are a & b and the hypotenuse is c.
a)a = 12 & c = 25b)a = x+2, b = x, c =
W.2Are 8, 12, and 17 lengths of a right triangle? Why?
W.3What is the value of y, if x = 3?
a)≈ 2.24
b)≈ 3.61
c)4
d)≈ 4.12
e)5
Find the distance between:
(1, 1) and (6, 4).
Plot the points.
Construct a right triangle.
Use the Pythagorean Theorem.
Use (x1, y1) & (x2, y2) to find the Distance formula:
Ex.1Find the distance between:
a) (3,-1) & (4,0)b)(5,-1) & (-3,7)
Ex.2Find the exact lengths of each side of quadrilateral EFGH. Then find the perimeter to the nearest tenth.
E: (-1, 5) F: (4, 3)G: (3, -2)H: (-2, -2)
Find the midpoint of
(-9, 4) and (-3, -2).
Plot the points and find
the point in the middle.
Midpoint formula:
Ex.3Find the midpoint between:
a)(5,-1) & (-3,7)b)(-8, 3) & (-3, -4)
Ex.4A circle is drawn on a coordinate plane. The endpoints of a diameter are (4, -3) and (-3, 5). What is the center of the circle?
Ex.5Are (-2,3), (-1,1), and (2,3) the vertices of a right
triangle?
Ex.6If Y is the midpoint of XZ, which of the following
must be true?
I.YZ = XZII.XZ = 2XYIII. 2XY = XZ
a)I only
b)II only
c)III only
d)I & II
e)I & III
Notes #3-___
Date:______
11-4 Operations with Radical Expressions (600)
W.1Find the distance between the points.
a)(2,6) , (8, 13)b)(-1, 7) , (5, 10)
W.2Find the midpoint of each segment with the given endpoints.
a)A (4,-1), B (2,11)b)H (-5, 6), K (1,7)
W.3Simplify each radical expression.
a)b)c)
Like radicals have the same .
and
Unlike radicals do not have the same .
and
Ex.1Combine the like radicals.
a)b)
Ex.2Simplify to combine like radicals.
a)b)
Ex.3Use the distributive property and simplify.
a)b)
What happens in general when you multiply conjugates together: ?
Ex.4Simplify using FOIL.
a)b)
Ex.5Rationalize the denominator.
a)b)
c)d)
Summary:
Ex.6The ratio length : width of a painting is approximately equal to the golden ratio . The length of the painting is 51 in. Find the exact width of the painting in simplest radical form. Then approximate width to the nearest inch.
Ex.7Is a solution of = 0?