1.3 Use midpoint and distance formulas You will find lengths of segments in the coordinate plane
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Transcript of 1.3 Use midpoint and distance formulas You will find lengths of segments in the coordinate plane
1.3 Use midpoint and distance formulas
You will find lengths of segments in the coordinate plane
Essential question: How do you find the distance and the midpoint between two points in the coordinate plane?
Warm-Up Exercises
In the skateboard design, VW bisects XY at point T, and XT = 39.9 cm. Find XY.
Skateboard
SOLUTION
EXAMPLE 1 Find segment lengths
Point T is the midpoint of XY . So, XT = TY = 39.9 cm.
XY = XT + TY= 39.9 + 39.9= 79.8 cm
Segment Addition PostulateSubstitute.
Add.
Warm-Up Exercises
SOLUTION
EXAMPLE 2 Use algebra with segment lengths
STEP 1 Write and solve an equation. Use the fact that VM = MW.
VM = MW4x – 1 = 3x + 3
x – 1 = 3x = 4
Write equation.
Substitute.
Subtract 3x from each side.Add 1 to each side.
Point M is the midpoint of VW . Find the length of VM .ALGEBRA
Warm-Up ExercisesEXAMPLE 2 Use algebra with segment lengths
STEP 2 Evaluate the expression for VM when x = 4.
VM = 4x – 1 = 4(4) – 1 = 15
So, the length of VM is 15.
Check: Because VM = MW, the length of MW should be 15. If you evaluate the expression for MW, you should find that MW = 15.
MW = 3x + 3 = 3(4) +3 = 15
Warm-Up ExercisesGUIDED PRACTICE for Examples 1 and 2
In Exercises 1 and 2, identify the segment bisectorof PQ . Then find PQ.
1.
343ANSWER MN;
Warm-Up ExercisesGUIDED PRACTICE for Examples 1 and 2
2.
In Exercises 1 and 2, identify the segment bisectorof PQ . Then find PQ.
line l ; 11 57
ANSWER
Warm-Up ExercisesEXAMPLE 3 Use the Midpoint Formula
a. FIND MIDPOINT The endpoints of RS are R(1,–3) and S(4, 2). Find the coordinates of the midpoint M.
Warm-Up ExercisesEXAMPLE 3 Use the Midpoint Formula
252
1 + 4 2
– 3 + 2 2 =, M , – 1M
The coordinates of the midpoint M are 1,–5
2 2
ANSWER
SOLUTION
a. FIND MIDPOINT Use the Midpoint Formula.
Warm-Up ExercisesEXAMPLE 3 Use the Midpoint Formula
b. FIND ENDPOINT The midpoint of JK is M(2, 1). One endpoint is J(1, 4). Find the coordinates of endpoint K.
Warm-Up ExercisesEXAMPLE 3 Use the Midpoint Formula
FIND ENDPOINT Let (x, y) be the coordinates of endpoint K. Use the Midpoint Formula.
STEP 1 Find x.
1+ x 22
=
1 + x = 4
x = 3
STEP 2 Find y.
4+ y 12
=
4 + y = 2
y = – 2
The coordinates of endpoint K are (3, – 2).ANSWER
SOLUTION
Warm-Up ExercisesGUIDED PRACTICE for Example 3
3. The endpoints of AB are A(1, 2) and B(7, 8). Find the coordinates of the midpoint M.
ANSWER (4,5)
ANSWER (– 6, – 8)
4. The midpoint of VW is M(– 1, – 2). One endpoint is W(4, 4). Find the coordinates of endpoint V.
Warm-Up Exercises
SOLUTION
EXAMPLE 4 Standardized Test Practice
Use the Distance Formula. You may find it helpful to draw a diagram.
Warm-Up ExercisesEXAMPLE 4 Standardized Test Practice
Distance Formula
Substitute.
Subtract.
Evaluate powers.
Add.
Use a calculator to approximate the square root.
(x – x ) + (y – y )2 2 2 2 1 1 RS =
[(4 – 2)] + [(–1) –3] 2 2=
(2) + (–4 )2 2=
4+16=
20=
The correct answer is C.ANSWER
4.47~=
Warm-Up ExercisesGUIDED PRACTICE for Example 4
5. In Example 4, does it matter which ordered pair you choose to substitute for (x , y ) and which ordered pair you choose to substitute for (x , y )? Explain.
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2
1
2
No, when squaring the differences in the coordinates, you get the same answer as long as you choose the x and y values from the same point.
SAMPLE ANSWER
Warm-Up ExercisesGUIDED PRACTICE for Example 4
6. What is the approximate length of AB , with endpoints A(–3, 2) and B(1, –4)?
6.1 units 7.2 units 8.5 units 10.0 units
BANSWER
Warm-Up ExercisesDaily Homework Quiz
1. AB bisects CD at E. If CE = in., Find CD.14
2
2. Point M is the midpoint of XY. Find XM.
ANSWER 17
12
4ANSWER in.
Warm-Up ExercisesDaily Homework Quiz
3. Point M is the midpoint of PQ with endpoints P(2, – 6 ) and Q(– 8, 0). Find the coordinates of M.
ANSWER (–3, –3)
ANSWER (3, –5)
4. The midpoint of GH is M(4, –1). One endpoint is G(5, 3) . Find the coordinates of H.
Warm-Up ExercisesDaily Homework Quiz
5. To find the distance between the swing and the sandbox in his backyard, Darren made a graph and found the coordinates of the swing to be (7, 2) and the coordinates of the sandbox to be (– 3, 8). Find the distance between the swing and the sandbox to the nearest tenth of a unit.
11.7 ANSWER
1.3 Use midpoint and distance formulas
You will find lengths of segments in the coordinate plane
Essential question: How do you find the distance and the midpoint between two points in the coordinate plane?
The midpoint of a segment is the point that divides the segment into two congruent parts.The length of a segment in the coordinate plane is the distance between its endpoints.
To find the distance between the points (a,b) and (c,d) use the distance formula d= (d-b)2+(c-a)2. To find the midpoint, use the midpoint formula (a+c, b+d) 2 2