Geometry – Section 5.1 – Notes and Examples – Perpendicular and Angle Bisectors

When a point is the __________ distance from two or more objects, the _____________ is said to be

________________ from the objects. Triangle ___________________ theorems can be used to prove

theorems about ___________________ points.

A _________ is a set of points that satisfies a given condition. The perpendicular ______________ of a

segment can be defined as the locus of points in a plane that are ___________________ from the

endpoints of the segment.

Remember that the _______________ between a __________ and a ________ is the length of the

______________________ segment from the point to the line.

1. The vertical line is the perpendicular bisector of the segment. 2. Place a point on the vertical line and label it A 3. Draw a segment from A to each endpoint. 4. Measure each segment 5. How do these measures compare?

1. The ray is bisecting the angle so show this by placing arc marks on the congruent angles. 2. Place a point on the angle bisector and label it B 3. Draw two segments from B, perpendicular to both sides. 4. Measure each segment. 5. How do these measures compare?

Problem 1 Find 𝑴𝑵.

Problem 2 Find 𝑩𝑪.

Problem 3 Find 𝑻𝑼.

Problem 4 Find 𝑩𝑪.

Problem 5 Find the 𝒎∠𝑬𝑭𝑯, given that 𝒎∠𝑬𝑭𝑮 = 𝟓𝟎°.

Problem 6 Find the 𝒎∠𝑴𝑲𝑳.

Problem 7

Given that 𝒀𝑾⃗⃗⃗⃗⃗⃗ ⃗ bisects ∠𝑿𝒀𝒁 and 𝑾𝒁 = 𝟑. 𝟎𝟓, find 𝑾𝑿.

Problem 8 Given that 𝒎∠𝑾𝒀𝒁 = 𝟔𝟑°, 𝑿𝑾 = 𝟓. 𝟕, and 𝒁𝑾 = 𝟓. 𝟕, find 𝒎∠𝑿𝒀𝒁.

Problem 9 Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints

𝑪(𝟔, – 𝟓) and 𝑫(𝟏𝟎, 𝟏).

Problem 10 Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints P(5, 2) and Q(1, –4).

Geometry – Section 5.2 – Notes and Examples – Bisectors of Triangles

Since a triangle has _________ sides, it has three ________________________ bisectors.

The _____________________ bisector of a ________ of a triangle does not always ________ through

the opposite __________.

When __________ or more lines intersect at ______ point, the lines are said to be ________________.

The __________ of _____________________ is the point where they ______________. In the

construction, you saw that the __________ perpendicular bisectors of a triangle are

________________. This point of concurrency is the _______________________ of the ____________.

The _____________________ can be ____________ the triangle, _____________ the triangle, or ____

the triangle.

The ______________________ of 𝛥𝐴𝐵𝐶 is the ___________ of its

circumscribed __________. A circle that contains all the vertices of a

polygon is ______________________ about the _____________.

A triangle has ___________ angles, so it has three angle ________________. The angle bisectors of a

triangle are also __________________. This point of concurrency is the _____________ of the

______________ .

The distance between a point and a line is the length of the perpendicular segment from the

point to the line.

Unlike the ______________________, the _____________ is always ____________ the triangle.

The ______________ is the center of the triangle’s inscribed circle. A

circle _______________ in a polygon ________________ each line

that contains a side of the polygon at exactly ______ point.

Problem 1 𝑫𝑮̅̅ ̅̅ , 𝑬𝑮̅̅ ̅̅ , and 𝑭𝑮̅̅ ̅̅ are the perpendicular bisectors of 𝑩𝑪. Find 𝑮𝑪.

Problem 2 𝑲𝒁̅̅ ̅̅ and 𝑴𝒁̅̅ ̅̅ ̅ are perpendicular bisectors of 𝑮𝑯𝑱. Find 𝑮𝑲, 𝑮𝑴, and 𝑱𝒁.

Problem 3 Find the circumcenter of 𝑯𝑱𝑲 with vertices 𝑯(𝟎, 𝟎), 𝑱(𝟏𝟎, 𝟎), and 𝑲(𝟎, 𝟔).

Problem 4 Find the circumcenter of 𝑮 𝑯 with vertices

𝑮(𝟎, – ), (𝟎, 𝟎), and 𝑯( , 𝟎).

Problem 5 A city planner wants to build a new library between a school, a post office, and a hospital. Draw a sketch to show where the library should be placed so it is the same distance from all three buildings.

Problem 6 Lee’s job requires him to travel to X, Y, and Z. Draw a sketch to show where he should buy a home so it is the same distance from all three places.

Problem 7 𝑴 ̅̅ ̅̅ ̅ and 𝑳 ̅̅ ̅̅ are angle bisectors of 𝑳𝑴𝑵. Find the distance from to 𝑴𝑵̅̅ ̅̅ ̅.

Problem 8 𝑴 ̅̅ ̅̅ ̅ and 𝑳 ̅̅ ̅̅ are angle bisectors of 𝑳𝑴𝑵. Find the 𝒎∠ 𝑴𝑵.

Problem 9 𝑿̅̅ ̅̅ and 𝑿̅̅ ̅̅ are angle bisectors of . Find the distance from 𝑿 to ̅̅ ̅̅ .

Problem 10 𝑿̅̅ ̅̅ and 𝑿̅̅ ̅̅ are angle bisectors of . Find the ∠ 𝑿 .

Geometry – Section 5.3 – Notes and Examples – Medians and Altitudes of Triangles

A __________ of a triangle is a ______________ whose

endpoints are a ___________ of the triangle and the

_______________ of the _____________ side.

Every triangle has __________ medians, and the

medians are ___________________.

The point of ___________________ of the ____________ of a triangle is the _____________ of the

triangle. The centroid is always _____________ the triangle. The centroid is also called the __________

___ ___________ because it is the point where a ___________________ region will _______________.

An ______________ of a ______________ is a perpendicular

____________ from a ____________ to the line containing the

________________ side. Every triangle has __________ altitudes. An

altitude can be __________, ____________, or ____ the triangle.

In 𝛥𝑄𝑅𝑆 to the right, ______________ 𝑄𝑌̅̅ ̅̅ is inside the triangle, but

𝑅𝑋̅̅ ̅̅ and 𝑆𝑍̅̅̅̅ are not. Notice that the lines containing the altitudes are

__________________ at 𝑃. This point of concurrency is the

____________________ of the triangle.

Geometry – Section 5.4 – Notes and Examples – The Triangle

Midsegment Theorem

A _________________ of a triangle is a segment that ________ the _______________ of two

_________ of the triangle. Every triangle has ____________ midsegments, which form the

____________________ ____________.

Problem 1 Find 𝑩𝑫 and the 𝒎∠𝑪𝑩𝑫.

Problem 2 Find 𝑱𝑳, 𝑴 and the 𝒎∠𝑴𝑳𝑲

Problem 3 Find .

Problem 4 𝑿𝒀𝒁 is the midsegment triangle of 𝑾𝑼 . What is the perimeter of 𝑿𝒀𝒁?

Problem 5

The vertices of 𝑿𝒀𝒁 are 𝑿(–𝟏, ), 𝒀( , ), and 𝒁(𝟑, – ). 𝑴 and 𝑵 are the midpoints of 𝑿𝒁̅̅ ̅̅ and 𝒀𝒁̅̅ ̅̅ .

Show that 𝑴𝑵̅̅ ̅̅ ̅ 𝑿𝒀̅̅ ̅̅ and 𝑴𝑵 =𝟏

𝑿𝒀.

Problem 6

The vertices of 𝑻 are (–𝟕, 𝟎), (–𝟑, 𝟔), and 𝑻( , ). 𝑴 is the midpoint of 𝑻̅̅ ̅̅ , and 𝑵 is the

midpoint of 𝑻̅̅̅̅ . Show that 𝑴𝑵̅̅ ̅̅ ̅ ̅̅ ̅̅ and 𝑴𝑵 =𝟏

.

_____________.

Problem 1 In 𝑳𝑴𝑵, 𝑳 = 𝟏 and = . Find 𝑳 and N .

Problem 2 In 𝑱𝑲𝑳, 𝒁𝑾 = 𝟕, and 𝑳𝑿 = . 𝟏. Find 𝑲𝑾 and 𝑳𝒁.

Problem 3 In 𝑩𝑪, 𝑬 = 𝟏 , 𝑫𝑮 = 𝟕, and 𝑩𝑮 = . Find the lengths of 𝑮, 𝑮𝑪, and 𝑮𝑭.

Problem 4 Find the average of the x-coordinates and the average of the y-coordinates of the vertices of ∆PQR. Make a conjecture about the centroid of a triangle.