In a triangle EFG, side EF has length 8 and FG has length 10.

1

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (A) Two of the possible lengths of side EG are 3
  and 16. Draw a triangle EFG with side EG of
  length 3. Draw a second triangle EFG with side
  EG of length 16. For each triangle label all of
  the vertices and the lengths of all of the sides.
  
• (B) The length of EG could not be 1 or 20.
  Explain why not. Draw figures to support your
  explanations.
• (C) If triangle EFG is a right triangle, what
  are the two possible lengths of side EG? Draw
  the two right triangles. For each triangle,
  label all vertices and the length of all sides.
  Indicate the right angle.

• PRAXIS - WI
• Kosiak
• Kaufmann
• Stracke

Triangle Constructed Response
2
Standards

• Praxis
• II. (MSM) Geometry and Measurement
• Apply the Pythagorean theorem to solve problems.
• Solve problems using the relationships among the
  parts of triangles such as sides and angles.

• WMAS
• C. Geometry
• C.8.1 Describe special and complex two- and
  three-dimensional figures (e.g., rhombus,
  polyhedron, cylinder) and their component parts
  (e.g., base, altitude, and slant height) by
  naming, defining, and giving examples and drawing
  and constructing physical models to
  specifications
• D. Measurement
• D.8.4 Determine measurements indirectly using
  estimation, geometric formulas to derive lengths,
  the Pythagorean relationship, and geometric
  relationships and properties for angle size

3
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (A) Two of the possible lengths of side EG are 3
  and 16. Draw a triangle EFG with side EG of
  length 3. Draw a second triangle EFG with side
  EG of length 16. For each triangle label all of
  the vertices and the lengths of all of the sides.

To construct triangle EFG, draw line segment EF
with length 8 and line segment FG with length 10.
Triangle Constructed Response
4
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (A) Two of the possible lengths of side EG are 3
  and 16. Draw a triangle EFG with side EG of
  length 3. Draw a second triangle EFG with side
  EG of length 16. For each triangle label all of
  the vertices and the lengths of all of the sides.

To construct triangle EFG, create vertex F by
joining the appropriate endpoints of the two line
segments.
Triangle Constructed Response
5
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (A) Two of the possible lengths of side EG are 3
  and 16. Draw a triangle EFG with side EG of
  length 3. Draw a second triangle EFG with side
  EG of length 16. For each triangle label all of
  the vertices and the lengths of all of the sides.

To construct the first triangle EFG, draw line
segment EG with length 3
Triangle Constructed Response
6
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (A) Two of the possible lengths of side EG are 3
  and 16. Draw a triangle EFG with side EG of
  length 3. Draw a second triangle EFG with side
  EG of length 16. For each triangle label all of
  the vertices and the lengths of all of the sides.

In a triangle, the angle opposite the longest
side must be the largest angle. Likewise, the
angle opposite the smallest side must be the
smallest angle.
Triangle Constructed Response
7
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (A) Two of the possible lengths of side EG are 3
  and 16. Draw a triangle EFG with side EG of
  length 3. Draw a second triangle EFG with side
  EG of length 16. For each triangle label all of
  the vertices and the lengths of all of the sides.

In triangle EFG, the angle at vertex E must be
the largest angle since it is opposite the
largest side. The angle at vertex F must be an
acute angle, because it is opposite the smallest
side.
Triangle Constructed Response
8
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (A) Two of the possible lengths of side EG are 3
  and 16. Draw a triangle EFG with side EG of
  length 3. Draw a second triangle EFG with side
  EG of length 16. For each triangle label all of
  the vertices and the lengths of all of the sides.

In triangle EFG, the angle at vertex E must be
the largest angle since it is opposite the
largest side. The angle at vertex F must be an
acute angle, because it is opposite the smallest
side.
Triangle EFG
8
10
3
Triangle Constructed Response
9
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (A) Two of the possible lengths of side EG are 3
  and 16. Draw a triangle EFG with side EG of
  length 3. Draw a second triangle EFG with side
  EG of length 16. For each triangle label all of
  the vertices and the lengths of all of the sides.

To construct the second triangle EFG, again
create vertex F by joining the appropriate
endpoints of the two line segments.
Triangle Constructed Response
10
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (A) Two of the possible lengths of side EG are 3
  and 16. Draw a triangle EFG with side EG of
  length 3. Draw a second triangle EFG with side
  EG of length 16. For each triangle label all of
  the vertices and the lengths of all of the sides.

Now draw line segment EG with length 16.
Triangle Constructed Response
11
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (A) Two of the possible lengths of side EG are 3
  and 16. Draw a triangle EFG with side EG of
  length 3. Draw a second triangle EFG with side
  EG of length 16. For each triangle label all of
  the vertices and the lengths of all of the sides.

In triangle EFG, the angle at vertex F must be
the largest angle since it is opposite the
largest side.
Triangle EFG
10
8
16
Triangle Constructed Response
12
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (B) The length of EG could not be 1 or 20.
  Explain why not. Draw figures to support your
  explanations.

In any triangle the sum of two shorter sides of
the triangle must always be greater than the
longest side.
For example, in ? EFG 8 3 gt 10
8
10
3
Triangle Constructed Response
13
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (B) The length of EG could not be 1 or 20.
  Explain why not. Draw figures to support your
  explanations.

If length EG was 1, the sum of the lengths of EF
and EG would be 9. This length is less than the
FG, which is 10.
8
8 1 lt 10
1
10
Triangle Constructed Response
14
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (B) The length of EG could not be 1 or 20.
  Explain why not. Draw figures to support your
  explanations.

If length EG was 1, the sum of the lengths of EF
and EG would be 9. This length is less than the
FG, which is 10.
8 1 lt 10
8
1
10
Triangle Constructed Response
15
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (B) The length of EG could not be 1 or 20.
  Explain why not. Draw figures to support your
  explanations.

If length EG was 1, the sum of the lengths of EF
and EG would be 9. This length is less than the
FG, which is 10.
Therefore, no triangle can be constructed with
side lengths of 1, 8, and 10
8
10
1
Triangle Constructed Response
16
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (B) The length of EG could not be 1 or 20.
  Explain why not. Draw figures to support your
  explanations.

If EG had a length of 20, EF had length 8, and FG
had a length of 10, then the sum of the two
smallest lengths is 8 10 18 This length is
less than the longest side EG, which is 20.
10
8
20
Triangle Constructed Response
17
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (B) The length of EG could not be 1 or 20.
  Explain why not. Draw figures to support your
  explanations.

Since 8 10 lt 20, no triangle can be
constructed with side lengths of 8, 10, and 20.
8
10
20
Triangle Constructed Response
18
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (C) If triangle EFG is a right triangle, what
  are the two possible lengths of side EG? Draw
  the two right triangles. For each triangle,
  label all vertices and the length of all sides.
  Indicate the right angle.

In a right triangle, the longest side is called
the hypotenuse. The hypotenuse is always the
side opposite the right angle.
If EF has length 8 and FG has length 10, then
there are two different right triangles that can
be constructed
Triangle Constructed Response
19
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (C) If triangle EFG is a right triangle, what
  are the two possible lengths of side EG? Draw
  the two right triangles. For each triangle,
  label all vertices and the length of all sides.
  Indicate the right angle.

In the first right triangle, we can choose FG to
be the hypotenuse. Under this condition, the
right angle is at vertex E.
8
10
8
10
?
Triangle Constructed Response
20
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (C) If triangle EFG is a right triangle, what
  are the two possible lengths of side EG? Draw
  the two right triangles. For each triangle,
  label all vertices and the length of all sides.
  Indicate the right angle.

To find the value of the missing side length, we
will use Pythagoreans Theorem which states that
the sum of the squares of the legs is equal to
the square of the hypotenuse.
c
a
b
Triangle Constructed Response
21
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (C) If triangle EFG is a right triangle, what
  are the two possible lengths of side EG? Draw
  the two right triangles. For each triangle,
  label all vertices and the length of all sides.
  Indicate the right angle.

c2
a2
For this right triangle,
b2
a2
b2
c2


Triangle Constructed Response
22
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (C) If triangle EFG is a right triangle, what
  are the two possible lengths of side EG? Draw
  the two right triangles. For each triangle,
  label all vertices and the length of all sides.
  Indicate the right angle.

If FG, which has length 10, is the hypotenuse,
then

82
b2
102

64 b2 100
b2 100 - 64
10
b2 36
8
b 6
b
Triangle Constructed Response
23
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (C) If triangle EFG is a right triangle, what
  are the two possible lengths of side EG? Draw
  the two right triangles. For each triangle,
  label all vertices and the length of all sides.
  Indicate the right angle.

If ? EFG is a right triangle, then the length of
EG could be 6.
10
8
6
Triangle Constructed Response
24
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (C) If triangle EFG is a right triangle, what
  are the two possible lengths of side EG? Draw
  the two right triangles. For each triangle,
  label all vertices and the length of all sides.
  Indicate the right angle.

In the second case, if ? EFG is a right triangle,
we can choose EG to be the hypotenuse. Under
this condition, the right angle is at vertex F.
10
8
8
c
10
Triangle Constructed Response
25
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (C) If triangle EFG is a right triangle, what
  are the two possible lengths of side EG? Draw
  the two right triangles. For each triangle,
  label all vertices and the length of all sides.
  Indicate the right angle.

In the right triangle ? EFG, if EG is the
hypotenuse, then
10
102
82
c2


64 100 c2
8
164 c2
c
12.8 c
Triangle Constructed Response
26
Tutorial

• In a triangle EFG, side EF has length 8 and FG
  has length 10.
• (C) If triangle EFG is a right triangle, what
  are the two possible lengths of side EG? Draw
  the two right triangles. For each triangle,
  label all vertices and the length of all sides.
  Indicate the right angle.

Returning to the original problem, if ? EFG is a
right triangle, then the length of EG could be
or 6.
10
10
8
8
6
Triangle Constructed Response
27

• Sally wanted construct a triangle with two of the
  sides
• having lengths 7 and 17. Which of the following
  side lengths
• is not a possible value for the length of the
  third side?
• 7
• 11
• 13
• 23

Triangle Constructed Response
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• 2. In ?ABC, AB has a length of 8, BC has a length
  of 14, and
• AC has a length of 9. Which of the vertices A,
  B, or C must
• be the largest angle?
• Vertex A
• Vertex B
• Vertex C
• Can not be determine from the given information.

Triangle Constructed Response
29

• 3. A scalene triangle has sides of x, 12, and 15.
  If the sides
• are listed in order from smallest to largest,
  what are
• all of the possible integer values for x?
• 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 11
• 3, 4, 5, 6, 7, 8, 9, 10, and 11
• 4, 5, 6, 7, 8, 9, 10, and 11
• 7, 8, 9, 10, and 11

Triangle Constructed Response
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• 4. A scalene triangle has sides of x, 12, and
  15. If the sides
• are listed in order from smallest to largest,
  determine the
• value for x such that the triangle is a right
  triangle.
• of x?
• 3
• 9
• about 19.2

Triangle Constructed Response
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• 5. In ? EFG below, vertex F is an obtuse angle.
  Determine
• the smallest possible integer values for the
  length of EG.
• 5 cm
• 6cm
• 18 cm
• 28 cm

17 cm
12 cm
Triangle Constructed Response
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• 6. In ? EFG below, vertex F is an obtuse angle.
  Determine
• the largest possible integer values for the
  length of EG.
• 6cm
• 18 cm
• 28 cm
• 29 cm

17 cm
12 cm
Triangle Constructed Response
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• 7. A 10-foot ladder is placed against a wall with
  its base
• 6-feet from the wall. How high above the ground
  is the top
• ladder?
• 6 feet
• 9 feet
• 10 feet
• 13 feet

Triangle Constructed Response
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• 8. Given the following sets of numbers, which
  set(s) could
• represent the side lengths of a right triangle?
• 12, 13, 5 II. 6, 4, 5 III. 17, 8, 15
  IV. 11, 13, 7
• I only
• III and IV only
• II and III only
• I and III only

Triangle Constructed Response
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9. Given the lengths 2cm, 5cm, and 7cm, is it
possible to construct a triangle with the given
lengths? Explain why or why not.
Triangle Constructed Response
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• In the isosceles triangle CAT below, the base,
  CA, is
• 10 cm. What is smallest possible integer length
  of the
• two congruent sides AT and CT?
• a) 5 cm
• b) 6 cm
• c) 8 cm
• d) 10 cm

T
10 cm
A
C
Triangle Constructed Response
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Describe the Hint and the Sandbox (in detail).
The sandbox contains fixed side lengths EF (8)
and FG (10) that are connected at vertex F.
Students can select an integer side length value
of EG and then determine if a triangle can be
made by increasing/decreasing the size of vertex
F and fitting in side EG.
In a triangle, the angle opposite the longest
side must be the largest angle.
Triangle Constructed Response