NJGPA Applying Geometric Theorems 3 — Questions and Answers
Question 1: A tangent and a secant are drawn from an external point to a circle. The tangent has length 8 and the external segment of the secant is 4. What is the length of the whole secant?
- 16 (Correct answer)
- 12
- 20
- 8
Correct answer: 16
Tangent-secant theorem: tangent² = external segment × whole secant, so 64 = 4 × whole secant, giving 16.
Question 2: In a regular hexagon, what is the measure of each interior angle?
- 120° (Correct answer)
- 108°
- 135°
- 90°
Correct answer: 120°
Each interior angle of a regular hexagon = (6 − 2) × 180° ÷ 6 = 720° ÷ 6 = 120°.
Question 3: Line segment MN has midpoint P. If M is at (2, 4) and P is at (5, 7), what are the coordinates of N?
- (8, 10) (Correct answer)
- (3, 3)
- (7, 11)
- (6, 9)
Correct answer: (8, 10)
Using the midpoint formula in reverse: N = (2×5 − 2, 2×7 − 4) = (8, 10).
Question 4: In triangle XYZ, XY is parallel to a line segment connecting the midpoints of XZ and YZ. If XY = 14, what is the length of the midsegment?
- 7 (Correct answer)
- 14
- 28
- 21
Correct answer: 7
By the Triangle Midsegment Theorem, the midsegment is half the length of the parallel side: 14 ÷ 2 = 7.
Question 5: Angles A and B are vertical angles. If angle A = (3x + 10)° and angle B = (5x − 6)°, what is x?
- 8 (Correct answer)
- 6
- 4
- 10
Correct answer: 8
Vertical angles are congruent: 3x + 10 = 5x − 6, so 16 = 2x, giving x = 8.
Question 6: A rectangle has a diagonal of 13 cm and a width of 5 cm. What is its length?
- 12 cm (Correct answer)
- 8 cm
- 10 cm
- 11 cm
Correct answer: 12 cm
Using the Pythagorean Theorem: length = √(13² − 5²) = √(169 − 25) = √144 = 12 cm.
Question 7: Two triangles are similar. The sides of the smaller triangle are 3, 4, and 5. The longest side of the larger triangle is 20. What is the perimeter of the larger triangle?
- 48 (Correct answer)
- 36
- 40
- 60
Correct answer: 48
The scale factor is 20 ÷ 5 = 4, so the sides are 12, 16, and 20, giving a perimeter of 48.
NJGPA Applying Geometric Theorems 3
A tangent and a secant are drawn from an external point to a circle.
The tangent has length 8 and the external segment of the secant is 4.
What is the length of the whole secant?