Transcript 9.7 Verifying Characteristics of a Geometric figure

Warm Up Quiz
1.
2.
3.
4.
If the lengths of a right triangle are 5 and 10
what could the missing side be?
[A] 75 [B] 5 5 [C] 5 [D] 125
If the hypotenuse of a 30-60-90 triangle is 30,
what are the other two sides?
If the leg of a right-isosceles triangle is 9, what
is the hypotenuse?
Find the distance between (-9,5) and (-7,3)
Answers
1.
B
2. 15 3 15
3. 9 2
4.
2 2
9.7 Verifying
Characteristics of a
Geometric figure
Formulas you need to know:
1)
y2  y1
Slope formula m 
x2 x1
1)
2)
equal
Slopes of parallel lines are ______________
Slopes of perpendicular lines are _______________
Opposite reciprocals
2)
Midpoint formula
 x  x2 y1  y2 
midpoint   1
,

2
2


3)
Distance formula
D  ( x2  x1 )2  ( y2  y1 )2
Ex1: Without graphing determine if a triangle
is right, justify your answer.
 Sides are A(-2,-1) B ( 1,2) C ( -2,5)
 Think: A right
triangle needs a right angle
 Perpendicular lines form right angles
 Which formula tells me something about
perpendicular lines?
 SLOPE – find the slope of each side (3)
2  1 3
mAB 
 1
1  2 3
52 3
mBC 
  1
2  1 3
Slopes are opposite reciprocals, so the lines are perpendicular so
we have a right angle so that means the triangle is right 
Ex2: Classify the triangle by its sides
defined by the points A(-3,5) B(-1,10) C(2,8)

We are working with sides, what formula
will tell us the length of a side?
 The
distance formula.
AB  (1  3)  (10  5)
2
 2 5
2
 29
2
2
BC  (2  1)2  (8  10)2
 (3)   2 
2
 13
2
AC  (2  3)2  (8  5)2
 52  32
 34
No sides are the same length therefore it is a
scalene triangle.
Justify the quad with sides A (3,1), B(1,5)
C(9,9), D( 11,5)is a rectangle B

C
A
D
Step 1: think what is the definition of a rectangle?
 A parallelogram with 4 right angles and opposite sides
are congruent
 I need to find the slope of each side.
95 4 1
5 1 4
2
mBC 
 
mAB 


9 1 8 2
1  3 2
1
5  9 4
2
mCD 
 
11  9 2
1
1  5 4 1
mDA 
 
3  11 8 2
DA and BC slopes are the same, and AB and CD slopes are the same so
opposite sides are parallel so it is a parallelogram. The slopes of AB and
AD are opposite reciprocals as are the slopes of BC and CD so the angles
are right. Now check to see if the opposite sides are congruent.
Justify the quad with sides A (3,1), B(1,5)
C(9,9), D( 11,5)is a rectangle
Use the distance formula to find the length of each side.
AB  (3  1) 2  (1  5) 2
CD  (9  11)  (9  5)
2
2
CB  (9  1) 2  (9  5) 2
AD  (3  11)  (1  5)
2
2
AB and CD are the same length, and CB and AD are the
same length. Since the angles are right angles, opposite
sides are both parallel and congruent, this figure is a
rectangle.
Review/ summary:
Start with definition of shape you are trying
to prove
 Use the formulas
 Answer the question in a complete
sentence.

Homework: 9.7 worksheet