Segments & Midpoints
A and B are at −8 and 5 on a number line. Find AB. Show your calculation.
AB = 13 units.
AB = |5 − (−8)| = |13| = 13 units.
Find the distance between P(−7, 4) and Q(5, 4). Show your calculation.
PQ = 12 units.
The y-coordinates match, so the segment is horizontal.
PQ = |5 − (−7)| = 12 units.
An angle is complementary to a 34° angle. Find its measure. Show your calculation.
56°.
Complementary angles add to 90°.
90° − 34° = 56°.
Two parallel lines are cut by a transversal. A pair of corresponding angles contains a 68° angle. Find the other angle and explain the relationship used.
68°.
Corresponding angles are congruent when the lines are parallel.
The other angle therefore also measures 68°.
A pentagon has side lengths 6 m, 8 m, 7 m, 10 m, and 9 m. Find its perimeter. Show your calculation.
P = 40 m.
Add every boundary side once.
P = 6 + 8 + 7 + 10 + 9 = 40 m.
Find the midpoint of A(−8, 3) and B(4, 9). Show substitution into the midpoint formula.
M(−2, 6).
M = ((−8 + 4)/2, (3 + 9)/2)
M = (−4/2, 12/2) = (−2, 6).
Find the distance between A(1, −2) and B(7, 6). Show substitution into the distance formula.
AB = 10 units.
AB = √[(7 − 1)² + (6 − (−2))²]
AB = √(6² + 8²) = √100 = 10 units.
Ray OB lies inside ∠AOC. If m∠AOC = 127° and m∠AOB = 48°, find m∠BOC. Show your equation.
m∠BOC = 79°.
m∠AOB + m∠BOC = m∠AOC
48° + m∠BOC = 127°
m∠BOC = 127° − 48° = 79°.
Two parallel lines are cut by a transversal. ∠A and ∠B are alternate interior angles, and m∠A = 112°. ∠B and ∠C form a linear pair. Find m∠B and m∠C. Show your reasoning.
m∠B = 112°; m∠C = 68°.
Alternate interior angles are congruent, so m∠B = 112°.
A linear pair sums to 180°.
m∠C = 180° − 112° = 68°.
A regular octagon has a perimeter of 100 cm. Find the length of one side. Show your equation.
One side = 12.5 cm.
A regular octagon has 8 congruent sides.
8s = 100 → s = 100/8 = 12.5 cm.
B is between A and C. AB = 3x + 4, BC = 2x − 1, and AC = 38. Find x, AB, and BC. Show your equation.
x = 7; AB = 25; BC = 13.
AB + BC = AC
(3x + 4) + (2x − 1) = 38
5x + 3 = 38 → x = 7
AB = 3(7) + 4 = 25; BC = 2(7) − 1 = 13.
Check: 25 + 13 = 38.
Find the exact distance between C(−3, 2) and D(3, 5). Simplify the radical and show your work.
CD = 3√5 units.
CD = √[(3 − (−3))² + (5 − 2)²]
CD = √(36 + 9) = √45
CD = √(9 × 5) = 3√5 units.
Two vertical angles measure (5x − 7)° and (3x + 25)°. Find x and the measure of each angle. Show your equation.
x = 16; each angle is 73°.
Vertical angles are congruent.
5x − 7 = 3x + 25
2x = 32 → x = 16
5(16) − 7 = 73° and 3(16) + 25 = 73°.
Two parallel lines are cut by a transversal. Corresponding angles measure (4x + 12)° and (6x − 18)°. Find x and both angle measures. Show your equation.
x = 15; both angles measure 72°.
Corresponding angles are congruent.
4x + 12 = 6x − 18
30 = 2x → x = 15
4(15) + 12 = 72°; 6(15) − 18 = 72°.
A rectangle has length 3x + 2 cm and width x − 1 cm. Its perimeter is 50 cm. Find x and both dimensions. Show your equation and work.
x = 6; length = 20 cm; width = 5 cm.
2(3x + 2) + 2(x − 1) = 50
8x + 2 = 50 → x = 6
Length = 3(6) + 2 = 20 cm; width = 6 − 1 = 5 cm.
Check: 2(20) + 2(5) = 50 cm.
M is the midpoint of AB. AM = 3x + 2 and MB = 5x − 10. Find x and AB. Show your equation and calculations.
x = 6; AB = 40.
Midpoint means AM = MB.
3x + 2 = 5x − 10
12 = 2x → x = 6
AM = 20 and MB = 20.
AB = 20 + 20 = 40.
A(2, −1) and B(x, 11) are 13 units apart. Find ALL possible values of x. Show your equation and work.
x = 7 or x = −3.
13² = (x − 2)² + (11 − (−1))²
169 = (x − 2)² + 144
(x − 2)² = 25 → x − 2 = ±5
x = 7 or x = −3.
Two angles form a linear pair. Their measures are (4x + 8)° and (2x + 28)°. Find x and both angle measures. Show your work.
x = 24; the angles are 104° and 76°.
A linear pair sums to 180°.
(4x + 8) + (2x + 28) = 180
6x + 36 = 180 → x = 24
4(24) + 8 = 104°; 2(24) + 28 = 76°.
Two parallel lines are cut by a transversal. Same-side interior angles measure (3x + 15)° and (5x + 5)°. Find x and both angle measures. Show your equation.
x = 20; the angles are 75° and 105°.
Same-side interior angles are supplementary.
(3x + 15) + (5x + 5) = 180
8x + 20 = 180 → x = 20
3(20) + 15 = 75°; 5(20) + 5 = 105°.
Triangle PQR has P(−2, 1), Q(4, 1), and R(4, 9). Find its perimeter. Show how you find each side length.
P = 24 units.
PQ = |4 − (−2)| = 6
QR = |9 − 1| = 8
PR = √(6² + 8²) = √100 = 10
Perimeter = 6 + 8 + 10 = 24 units.
A(−5, 7) is one endpoint of AB. The midpoint is M(2, −1). Find the other endpoint B. Show work for both coordinates.
B(9, −9).
(−5 + x)/2 = 2 → −5 + x = 4 → x = 9
(7 + y)/2 = −1 → 7 + y = −2 → y = −9
Check: ((−5 + 9)/2, (7 − 9)/2) = (2, −1).
Triangle ABC has A(−4, 1), B(2, 9), and C(8, 1). Calculate all three side lengths and decide whether it is isosceles. Show your work.
AB = BC = 10; AC = 12 units. Yes, it is isosceles.
AB = √[(2 + 4)² + (9 − 1)²] = √100 = 10
BC = √[(8 − 2)² + (1 − 9)²] = √100 = 10
AC = |8 − (−4)| = 12
AB = BC, so two sides are congruent.
Ray BD bisects ∠ABC. m∠ABD = (3x + 4)° and m∠DBC = (5x − 12)°. Find x, m∠ABC, and the measure of an angle supplementary to ∠ABC. Show your work.
x = 8; m∠ABC = 56°; supplementary angle = 124°.
Bisector means equal halves: 3x + 4 = 5x − 12.
16 = 2x → x = 8
Each half is 3(8) + 4 = 28°.
m∠ABC = 28° + 28° = 56°.
Supplementary angle = 180° − 56° = 124°.
Two parallel lines are cut by a transversal. Same-side exterior angles ∠A and ∠B measure (7x − 9)° and (4x + 13)°. ∠C is vertical to ∠A. Find x and all three angle measures. Show your work.
x = 16; m∠A = 103°; m∠B = 77°; m∠C = 103°.
Same-side exterior angles are supplementary.
(7x − 9) + (4x + 13) = 180
11x + 4 = 180 → x = 16
m∠A = 7(16) − 9 = 103°; m∠B = 4(16) + 13 = 77°.
Vertical angles are congruent, so m∠C = 103°.
A triangular park has A(−6, 2). M(0, 2) is the midpoint of AB, and C is (6, 11). Find B, all three side lengths, and the total fencing needed. Each coordinate unit is 1 meter. Show your work.
B(6, 2); AB = 12 m; BC = 9 m; AC = 15 m; fencing = 36 m.
B = (2·0 − (−6), 2·2 − 2) = (6, 2).
AB = |6 − (−6)| = 12 m; BC = |11 − 2| = 9 m.
AC = √(12² + 9²) = √225 = 15 m.
Perimeter = 12 + 9 + 15 = 36 m.