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 M in XZ .
s s d. At what point do PR and TZ intersect? Explain. s s s s PR and TZ do not intersect. TZ lies in plane M but only point P of TZ lies in M. Find Measurements by Subtracting
Find FG. Assume that the figure is not drawn to scale. in. FG is the measure of . Point G is between F and H. Find F FG G H FG by subtracting GH from FH. FG + GH = FH Betweenness of points 15 in. 1 FG + 10 = 15 Substitution 4 1 1 1 1 FG +10 10 = 15 10 Subtract 10 from each side. 4 4 4 4 3 FG = 4 in. Simplify. 4
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 + NP Betweenness of points 60 60 = 6x - 7 + 2x + 3 Substitution 6x - 7 2x + 3 60 = 8x – 4 Simplify. M N P 64 = 8x Add 4 to each side. 8 = x Divide each side by 8. Now find MN. MN = 6x – 7 Given = 6(8) – 7 x = 8 MN = 41 Simplify.
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 5 Simplify.
Find Distance on a Coordinate Plane
Find the distance between A(4, 2) and B(-6, 4). 2 2 AB = x -x +y - y Distance Formula ( 2 1 ) ( 2 1 ) 2 2 = (-6- 4) +(4 - 2) (x1,y1) = (4, 2),(x2,y2) = (-6, 4) 2 2 = (-10) + ( 2) Simplify. = 104 Simplify. The distance from A to B is 104 units. You can use a calculator to find that 104 is approximately 10.20.
Check Graph the ordered pairs and apply the Pythagorean Theorem.
(AB)2 (BC)2 (AC)2 (AB)2 22 102 (AB)2 4 100 AB 104 Find the Midpoint in a Coordinate Plane
Find the coordinates of M, the midpoint of RS , for R(-3, -4) and S(5, 7).
骣x +x y + y 琪1 2 , 1 2 M = 琪 Midpoint Formula 桫 2 2 骣-3+ 5 -4 + 7 = 琪 , (x1, y1) = (-3, -4), (x2, y2) = (5, 7) 桫 2 2 骣2 3 骣 1 = 琪 , or 琪1,1 Simplify. 桫2 2 桫 2
Find the Coordinates of an Endpoint
Find the coordinates of P if M(3, 2) is the mid point of KP and K has the coordinates (1,-5). Step 1 Let P be (x1, y1) and K be (x2, y2) in the Midpoint Formula. 骣1+x - 5 + y M琪 2, 2 = M (3,2) 琪 (x1, y1) = (1, -5) 桫 2 2 Step 2 Write two equations to find the coordinates of P. 1 + x -5+ y 2 = 3 Midpoint Formula 2 = 2 Midpoint Formula 2 2 1 + x2 = 6 Multiply each side by 2. -5 + y2 = 4 Multiply each side by 2. x2 = 5 Subtract 1 from each side. y2 = 9 Add 5 to each side.
The coordinates of P are (5, 9).
Use Algebra to Find Measures Find the measure of QR if Q is the midpoint of PR . 4a + 27 R Because Q is the midpoint, you know that PQ = QR. Use this equation to find a value for a. -6a – 3 Q
PQ = QR Definition of midpoint P -6a - 3 = 4a + 27 PQ = -6a – 3, QR = 4a + 27 -3 = 10a + 27 Add 6a to each side. -30 = 10a Subtract 27 from each side. -3 = a Divide each side by 10. Now substitute -3 for a in the expression for PQ. QR = 4a + 27 Original measure. = 4(-3) + 27 a = -3 = -12 + 27 or 15 Simplify.
The measure of QR 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. F E Student 8, 6, HKJ, and JKC 1 Parking Science D b. Name the sides of 6. 2 Lab 5 3 KJ and or KJ and H K C G Office KC KG 8 6 4 E Gymnasium B c. What is another name for AJK? 9 10 7 J A 9, J, and KJA
d. Name a point in the interior of ECG. Point F. Measure and Classify Angles D M In the figure, ABD FHG. If mABD = 3x + 6 and mFHG = x + 26, find the measures of ABD and FHG. B 蠤ABD FHG Given G m ABD = m FHG Definition of congruent angles A 3x + 6 = x + 26 Substitution 3x = x + 20 Subtract 6 from each side. H 2x = 20 Subtract x from each side. x = 10 Divide each side by 2. F J Use the value of x to find the measure of one angle. m ABD = 3x + 6 Given = 3(10) + 6 x = 10 = 30 + 6 or 36 Simplify.
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 B H
Name an angle pair that satisfies each condition. a. two acute adjacent angles
C
A D G F 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 ALGEBRA Find x and y so that DG and BE are perpendicular. If DG BE , then mDFB = 90 and mGFE = 90. To find x, use BFC and DFC. m BFC + mD F C = mDFB Sum of parts = whole 2x + 4x = 90 Substitution 6x = 90 Combine like terms. x = 15 Divide each side by 6.
To find y, use GFE. m GFE = 90 Given 5y + 20 = 90 Substitution 5y = 70 Subtract 20 from each side. y = 14 Divide 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