Example 4 Interpret Drawings
Interpret Drawings
a.How many planes appear in this figure?
There are four planes: plane M, plane XYZ,
plane ZTY, and plane XTY
b.Name three points that are collinear.
Points Y, Q, and Z are collinear.
c.Name the intersection of plane XYZ and plane M.
Plane XYZ intersects plane Min.
d.At what point do and intersect? Explain.
and do not intersect. lies in plane M but only point P of lies in M.
Find Measurements by Subtracting
Find FG. Assume that the figure is not drawn to scale.
FG is the measure of . Point G is between F and H. Find FG by subtracting GH from FH.
FG + GH / = FH / Betweenness of pointsFG + / = 15 / Substitution
FG + / = / Subtract from each side.
FG / = in. / Simplify.
Write and Solve Equations to Find Measurements
Find the value of x and MN if N is between M and P, MP = 60, MN = 6x - 7, and NP = 2x + 3.
Draw a figure to represent this information. Then write and solve an equation relating the given measures.
MP= MN + NPBetweenness of points
60= 6x - 7 + 2x + 3Substitution
60 = 8x – 4Simplify.
64 = 8xAdd 4 to each side.
8 = xDivide each side by 8.
Now find MN.
MN = 6x – 7Given
= 6(8) – 7x = 8
MN = 41Simplify.
Find Distance on a Number Line
Use the number line to find QR.
The coordinates of Q and R are -2 and 3.
QR = |x2 - x1| Distance Formula
= |-2 - 3|x1 = -2and x2 = -3
= |-5| or 5Simplify.
Find Distance on a Coordinate Plane
Find the distance between A(4, 2) and B(-6, 4).
AB = Distance Formula
= (x1,y1) = (4, 2),(x2,y2) = (-6, 4)
= Simplify.
= Simplify.
The distance from A to B is units. You can use a calculator to find that is approximately 10.20.
Check
Graph the ordered pairs and apply the Pythagorean Theorem.
Find the Midpoint in a Coordinate Plane
Find thecoordinates of M, the midpoint of, for R(-3, -4) and S(5, 7).
M = Midpoint Formula
= (x1, y1) = (-3, -4), (x2, y2) = (5, 7)
= or Simplify.
Find the Coordinates of an Endpoint
Find the coordinates of P if M(3, 2) is the mid point of and K has the coordinates (1,-5).
Step 1 Let P be (x1, y1) and K be (x2, y2) in the Midpoint Formula.
(x1, y1) = (1, -5)
Step 2 Write two equations to find the coordinates of P.
= 3Midpoint Formula = 2Midpoint Formula
1 + x2 = 6Multiply each side by 2.-5 + y2 = 4Multiply each side by 2.
x2 = 5Subtract 1 from each side.y2 = 9Add 5 to each side.
The coordinates of P are (5, 9).
Use Algebra to Find Measures
Find the measure of if Q is the midpoint of .
Because Q is the midpoint, you know that PQ = QR. Use this
equation to find a value for a.
PQ = QRDefinition of midpoint
-6a- 3 = 4a + 27PQ = -6a – 3, QR = 4a + 27
-3 = 10a + 27Add 6a to each side.
-30 = 10aSubtract 27 from each side.
-3 = aDivide each side by 10.
Now substitute -3 for a in the expression for PQ.
QR = 4a + 27Original measure.
= 4(-3) + 27a = -3
= -12 + 27 or 15Simplify.
The measure of is 15.
You can check your answer by substitution -3 for a in the expression for PQ. It should also have a length of 15.
Angles and Their Parts
Use the map of a high school shown.
a.Name all angles that have K as a vertex.
8, 6, HKJ, and JKC
b.Name the sides of 6.
and or and
c.What is another name for AJK?
9, J, and KJA
d.Name a point in the interior of ECG.
Point F.
Measure and Classify Angles
In the figure, ABDFHG. If mABD = 3x + 6 and
mFHG = x + 26, find the measures of ABD and FHG.
Given
Definition of congruent angles
3x + 6 = x + 26Substitution
3x = x + 20Subtract 6 from each side.
2x = 20Subtract x from each side.
x = 10Divide each side by 2.
Use the value of x to find the measure of one angle.
= 3x+ 6Given
= 3(10) + 6x = 10
= 30 + 6 or 36Simplify.
The measures of ABD and FHG are 36.
You can check you solution by substituting 10 for x in the expression for FHG.
Identify Angles Pairs
Name an angle pair that satisfies each condition.
a.two acute adjacent angles
BCH, ACD, DCG, and FCG are acute angles.
ACD and DCG are acute adjacent angles, and FCG and DCG are acute adjacent angles.
b.two obtuse vertical angles
BCD and HCG are obtuse vertical angles.
Perpendicular Lines
ALGEBRAFind x and y so that and are
perpendicular.
If , then mDFB = 90 and mGFE = 90.
To find x, use BFC and DFC.
= mDFBSum of parts = whole
2x + 4x= 90Substitution
6x = 90Combine like terms.
x = 15Divide each side by 6.
To find y, use GFE.
= 90Given
5y + 20 = 90Substitution
5y = 70Subtract 20 from each side.
y = 14Divide each side by 5.
Interpret Figures
Determine whether each statement can
be assumed from the figure. Explain.
a.BFC and AFG are complementary.
No; they are congruent, but we do not know
anything about their exact measurements.
b.DFA and AFG are a linear pair.
Yes; they are adjacent angles whose noncommon
sides are opposite rays.
c.DFC and BFC are complementary.
Yes; there is a right angle symbol showing the
adjacent angles form a right angle