4-1 Classifying Triangles
5-1 Perpendicular and Angle Bisectors
HW: 3, 7, 13, 15, 17, 23, 27, 31
New Terms:
When a point is the same distance from two or more objects, the point is said to be __________ from the objects. Triangle congruence theorems can be used to prove theorems about equidistant points.
A _______ is a set of points that satisfies a given condition. The perpendicular bisector of a segment can be defined as the locus of points in a plane that are equidistant from the endpoints of the segment.
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Example 1:
Solve for x.
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Example 2:
Given that YW bisects (XYZ and
WZ = 3.05, find WX.
Example 3:
Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints C(6, –5) and D(10, 1).
Example 5:
Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints P(5, 2) and Q(1, –4).
5-2 Bisectors of Triangles
HW: 3-9 odd, 13-17 odd, 23-31 odd, 35
When three or more lines intersect at one point, the lines are said to be ___________. The ____________________ is the point where they intersect. In the construction, you saw that the three perpendicular bisectors of a triangle are concurrent. This point of concurrency is the __________ of the triangle.
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Triangle Types and their Circumcenters:
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The circumcenter of ΔABC is the center of its circumscribed circle. A circle that contains all the vertices of a polygon is ______________ about the polygon.
Example 1:
Find the circumcenter of ∆GOH with vertices G(0, –9), O(0, 0), and H(8, 0).
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Triangle Types and their Incenters:
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The incenter is the center of the triangle’s inscribed circle. A circle inscribed in a polygon intersects each line that contains a side of the polygon at exactly one point.
Example 2:
MP and LP are angle bisectors of ∆LMN. Find the distance from P to MN.
5-3 Medians and Altitudes of Triangles
HW: 3-6, 9, 12-15, 21-24, 28, 29-32
A __________ of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.
Every triangle has three medians, and the medians are concurrent.
The point of concurrency of the medians of a triangle is the _________ of the triangle:
- Always inside the triangle
- Also called the center of gravity because it is the point where the triangular region will balance.
Example 1:
In ∆LMN, RL = 21 and SQ =4. Find NQ.
Example 2:
A sculptor is shaping a triangular piece of iron that will balance on the point of a cone. At what coordinates will the triangular region balance?
An __________ of a triangle is a perpendicular segment from a vertex to the line containing the opposite side.
- Every triangle has three of them.
- They can be inside, outside, or on the triangle.
- The point of concurrency is the ____________ of the triangle.
- The height of the triangle is the same thing.
Example 3:
Find the orthocenter of ∆XYZ with vertices X(3, –2), Y(3, 6), and Z(7, 1).
FYI…
5-7 The Pythagorean Theorem
HW: p.352 2-14 even, 30-35
Example 1:
Find the value of x. Give your answer in simplest radical form.
Example 2:
According to the recommended safety ratio of 4:1, how high will a 30-foot ladder reach when placed against a wall? Round to the nearest inch.
Definition:
A set of three nonzero whole numbers a, b, and c such that a2 + b2 = c2 is called a ____________________.
Example 3:
Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain. (Is the missing side a whole number?)
|The converse of the Pythagorean Theorem gives you a way to tell |You can also use side lengths to classify a triangle as acute or |
|if a triangle is a right triangle when you know the side lengths.|obtuse. |
Simplified…
Remember: By the Triangle Inequality Theorem, the sum of any two side lengths of a triangle is greater than the third side length.
Example 4:
Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.
5, 7, 10
5, 8, 17
5-8 Applying Special Right Triangles
HW: p.360 1-11 odd, 13-15
Example 1:
Find the value of x. Give your answer in simplest radical form.
Example 2:
Find the value of x. Give your answer in simplest radical form.
Example 3:
Find the values of x and y. Give your answers in simplest radical form.
Example 4:
Find the values of x and y. Give your answers in simplest radical form.
Example 5:
An ornamental pin is in the shape of an equilateral triangle. The length of each side is 6 centimeters. Josh will attach the fastener to the back along AB. Will the fastener fit if it is 4 centimeters long?
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a2 + b2 = c2
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